Digital flowmeter

ABSTRACT

A control and measurement system for a coriolis flowmeter having a flowtube, a driver adapted to vibrate the flowtube, and a pair of sensors adapted to generate signals indicative of movement of the flowtube when it is being vibrated by the driver, wherein the sensors are positioned relative to one another so the signals from the sensors are indicative of a mass flow rate of fluid through the flowtube. A digital drive signal generator is adapted to generate a variable digital drive signal for controlling operation of the driver. The digital drive signal generator can be adapted to cause the driver to resist motion of the flowtube during a first time period and amplify motion of the flowtube during a second time period. The digital drive signal generator can also be adapted to initiate motion of the flowtube by sending one or more square wave signals to the driver.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.13/853,793 filed Mar. 29, 2013, titled DIGITAL FLOWMETER, which is aContinuation of U.S. patent application Ser. No. 13/198,313, filed Aug.4, 2011, titled DIGITAL FLOWMETER, now allowed, which is a continuationof U.S. patent application Ser. No. 11/930,936, filed Oct. 31, 2007,titled DIGITAL FLOWMETER, now U.S. Pat. No. 8,000,906, which is acontinuation of U.S. application Ser. No. 11/555,033, filed Oct. 31,2006, titled DIGITAL FLOWMETER, now U.S. Pat. No. 7,571,062, which is acontinuation of U.S. application Ser. No. 11/130,233, filed May 17,2005, now U.S. Pat. No. 7,136,761, titled DIGITAL FLOWMETER, which is acontinuation of U.S. application Ser. No. 10/637,620, filed Aug. 11,2003, now U.S. Pat. No. 6,917,887, titled DIGITAL FLOWMETER, which is acontinuation of U.S. application Ser. No. 09/931,057, filed Aug. 17,2001, now U.S. Pat. No. 6,754,594, titled DIGITAL FLOWMETER, which is acontinuation of U.S. application Ser. No. 09/111,739, filed Jul. 8,1998, now U.S. Pat. No. 6,311,136, titled DIGITAL FLOWMETER, whichclaims priority from U.S. Provisional Application No. 60/066,554, filedNov. 26, 1997, titled DIGITAL FLOWMETER, all of which are incorporatedby reference.

TECHNICAL FIELD

The invention relates to flowmeters.

BACKGROUND

Flowmeters provide information about materials being transferred througha conduit. For example, mass flowmeters provide a direct indication ofthe mass of material being transferred through a conduit. Similarly,density flowmeters, or densitometers, provide an indication of thedensity of material flowing through a conduit. Mass flowmeters also mayprovide an indication of the density of the material.

Coriolis-type mass flowmeters are based on the well-known Corioliseffect, in which material flowing through a rotating conduit becomes aradially traveling mass that is affected by a Coriolis force andtherefore experiences an acceleration. Many Coriolis-type massflowmeters induce a Coriolis force by sinusoidally oscillating a conduitabout a pivot axis orthogonal to the length of the conduit. In such massflowmeters, the Coriolis reaction force experienced by the travelingfluid mass is transferred to the conduit itself and is manifested as adeflection or offset of the conduit in the direction of the Coriolisforce vector in the plane of rotation.

Energy is supplied to the conduit by a driving mechanism that applies aperiodic force to oscillate the conduit. One type of driving mechanismis an electromechanical driver that imparts a force proportional to anapplied voltage. In an oscillating flowmeter, the applied voltage isperiodic, and is generally sinusoidal. The period of the input voltageis chosen so that the motion of the conduit matches a resonant mode ofvibration of the conduit. This reduces the energy needed to sustainoscillation. An oscillating flowmeter may use a feedback loop in which asensor signal that carries instantaneous frequency and phase informationrelated to oscillation of the conduit is amplified and fed back to theconduit using the electromechanical driver.

SUMMARY

The invention provides a digital flowmeter, such as a digital massflowmeter, that uses a control and measurement system to controloscillation of the conduit and to generate mass flow and densitymeasurements. Sensors connected to the conduit supply signals to thecontrol and measurement system. The control and measurement systemprocesses the signals to produce a measurement of mass flow and usesdigital signal processing to generate a signal for driving the conduit.The drive signal then is converted to a force that induces oscillationof the conduit.

The digital mass flowmeter provides a number of advantages overtraditional, analog approaches. From a control perspective, use ofdigital processing techniques permits the application of precise,sophisticated control algorithms that, relative to traditional analogapproaches, provide greater responsiveness, accuracy and adaptability.

The digital control system also permits the use of negative gain incontrolling oscillation of the conduit. Thus, drive signals that are180° out of phase with conduit oscillation may be used to reduce theamplitude of oscillation. The practical implications of this areimportant, particularly in high and variable damping situations where asudden drop in damping can cause an undesirable increase in theamplitude of oscillation. One example of a variable damping situation iswhen aeration occurs in the material flowing through the conduit.

The ability to provide negative feedback also is important when theamplitude of oscillation is controlled to a fixed setpoint that can bechanged under user control. With negative feedback, reductions in theoscillation setpoint can be implemented as quickly as increases in thesetpoint. By contrast, an analog meter that relies solely on positivefeedback must set the gain to zero and wait for system damping to reducethe amplitude to the reduced setpoint.

From a measurement perspective, the digital mass flowmeter can providehigh information bandwidth. For example, a digital measurement systemmay use analog-to-digital converters operating at eighteen bits ofprecision and sampling rates of 55 kHz. The digital measurement systemalso may use sophisticated algorithms to filter and process the data,and may do so starting with the raw data from the sensors and continuingto the final measurement data. This permits extremely high precision,such as, for example, phase precision to five nanoseconds per cycle.Digital processing starting with the raw sensor data also allows forextensions to existing measurement techniques to improve performance innon-ideal situations, such as by detecting and compensating fortime-varying amplitude, frequency, and zero offset.

The control and measurement improvements interact to provide furtherimprovements. For example, control of oscillation amplitude is dependentupon the quality of amplitude measurement. Under normal conditions, thedigital mass flowmeter may maintain oscillation to within twenty partsper million of the desired setpoint. Similarly, improved control has apositive benefit on measurement. Increasing the stability of oscillationwill improve measurement quality even for meters that do not require afixed amplitude of oscillation (i.e., a fixed setpoint). For example,with improved stability, assumptions used for the measurementcalculations are likely to be valid over a wider range of conditions.

The digital mass flowmeter also permits the integration of entirely newfunctionality (e.g., diagnostics) with the measurement and controlprocesses. For example, algorithms for detecting the presence of processaeration can be implemented with compensatory action occurring for bothmeasurement and control if aeration is detected.

Other advantages of the digital mass flowmeter result from the limitedamount of hardware employed, which makes the meter simple to construct,debug, and repair in production and in the field. Quick repairs in thefield for improved performance and to compensate for wear of themechanical components (e.g., loops, flanges, sensors and drivers) arepossible because the meter uses standardized hardware components thatmay be replaced with little difficulty, and because softwaremodifications may be made with relative ease. In addition, integrationof diagnostics, measurement, and control is simplified by the simplicityof the hardware and the level of functionality implemented in software.New functionality, such as low power components or components withimproved performance, can be integrated without a major redesign of theoverall control system.

In one general aspect, a digital flowmeter includes a vibratableconduit, a driver connected to the conduit and operable to impart motionto the conduit, and a sensor connected to the conduit and operable tosense the motion of the conduit. A control and measurement systemconnected to the driver and the sensor includes circuitry that receivesa sensor signal from the sensor, generates a drive signal based on thesensor signal using digital signal processing, supplies the drive signalto the driver, and generates a measurement of a property of materialflowing through the conduit based on the signal from the sensor.

Embodiments may include one or more of the following features. The metermay include a second sensor connected to the conduit and operable tosense the motion of the conduit. In this case, the control andmeasurement system is connected to the second sensor and receives asecond sensor signal from the second sensor, generates the drive signalbased on the first and second sensor signals, and generates themeasurement of the property of material flowing through the conduitbased on the first and second sensor signals. The control andmeasurement system may digitally combine the first and second sensorsignals and generate the drive signal based on the combination of thesensor signals.

The control and measurement system may generate different drive signalsfor the two drivers. The drive signals may have, for example, differentfrequencies or amplitudes.

The digital flowmeter also may include circuitry for measuring currentsupplied to the driver. The circuitry may include a resistor in serieswith the driver and an analog-to-digital converter in parallel with theresistor and configured to measure a voltage across the resistor, toconvert the measured voltage to a digital value, and to supply thedigital value to the control and measurement system.

The digital flowmeter also may include a first pressure sensor connectedto measure a first pressure at an inlet to the conduit and a secondpressure sensor connected to measure a second pressure at an outlet ofthe conduit. Analog-to-digital converters may be connected andconfigured to convert signals produced by the first pressure sensor andthe second pressure sensor to digital values and to supply the digitalvalues to the control and measurement system. Temperature sensors may beconnected to measure temperatures at the inlet and outlet of theconduit.

The control and measurement system may generate the measurement of theproperty by estimating a frequency of the first sensor signal,calculating a phase difference using the first sensor signal, andgenerating the measurement using the calculated phase difference. Thecontrol and measurement system may compensate for amplitude differencesin the sensor signals by adjusting the amplitude of one of the sensorsignals. For example, the control and measurement system may multiplythe amplitude of one of the sensor signals by a ratio of the amplitudesof the sensor signals.

When the sensor signal is generally periodic, the control andmeasurement system may process the sensor signal in sets. Each set mayinclude data for a complete cycle of the periodic sensor signal, andconsecutive sets may include data for overlapping cycles of the periodicsensor signal. The control and measurement system may estimate an endpoint of a cycle using a frequency of a previous cycle.

The control and measurement system may analyze data for a cycle todetermine whether the cycle merits further processing. For example, thesystem may determine that a cycle does not merit further processing whendata for the cycle does not conform to expected behavior for the data,where the expected behavior may be based on one or more parameters of aprevious cycle. In one implementation, the system determines that acycle does not merit further processing when a frequency for the cyclediffers from a frequency for the previous cycle by more than a thresholdamount. The system may determine whether the frequencies differ bycomparing values at certain points in the cycle to values that wouldoccur if the frequency for the cycle equaled the frequency for theprevious cycle.

The control and measurement system may determine a frequency of thesensor signal by detecting zero-crossings of the sensor signal andcounting samples between zero crossings. The system also may determine afrequency of the sensor signal using an iterative curve fittingtechnique.

The control and measurement system may determine an amplitude of thesensor signal using Fourier analysis, and may use the determinedamplitude in generating the drive signal.

The control and measurement system may determine a phase offset for eachsensor signal and may determine the phase difference by comparing thephase offsets. The system also may determine the phase difference usingFourier analysis. The control and measurement system may determine afrequency, amplitude and phase offset for each sensor signal, and mayscale the phase offsets to an average of the frequencies of the sensorsignals. The control and measurement system may calculate the phasedifference using multiple approaches and may select a result of one ofthe approaches as the calculated phase difference.

The control and measurement system may combine the sensor signals toproduce a combined signal and may generate the drive signal based on thecombined signal. For example, the control and measurement system may sumthe sensor signals to produce the combined signal and may generate thedrive signal by applying a gain to the combined signal.

In general, the control and measurement system may initiate motion ofthe conduit by using a first mode of signal generation to generate thedrive signal, and may sustain motion of the conduit using a second modeof signal generation to generate the drive signal. The first mode ofsignal generation may be synthesis of a periodic signal having a desiredproperty, such as a desired initial frequency of conduit vibration, andthe second mode of signal generation may use a feedback loop includingthe sensor signal.

In other instances, the first mode of signal generation may include useof a feedback loop including the sensor signal and the second mode ofsignal generation may include synthesis of a periodic signal having adesired property. For example, the control and measurement system maygenerate the drive signal by applying a large gain to the combinedsignal to initiate motion of the conduit and generating a periodicsignal having a phase and frequency based on a phase and frequency of asensor signal as the drive signal after motion has been initiated. Thedesired property of the synthesized signal may be a frequency and aphase corresponding to a frequency and a phase of the sensor signal.

The control and measurement system generates an adaptable, periodicdrive signal. For example, the meter may include positive and negativedirect current sources connected between the control and measurementsystem and the driver, and the control and measurement system maygenerate the drive signal by switching the current sources on and off atintervals having a phase and frequency based on the sensor signal. Thecontrol and measurement system may generate the drive signal bysynthesizing a sine wave having a property corresponding to a propertyof the sensor signal, such as a phase and a frequency corresponding to aphase and a frequency of the sensor signal.

The control and measurement system may digitally generate a gain for usein generating the drive signal based on one or more properties of thesensor signal. For example, the control and measurement system maydigitally generate the gain based on an amplitude of the sensor signal.

The driver may be operable to impart an oscillating motion to theconduit. The control and measurement system also may digitally implementa PI control algorithm to regulate the amplitude of conduit oscillation.The control and measurement system also may digitally generate the drivesignal based on the sensor signal so as to maintain an amplitude ofoscillation of the conduit at a user-controlled value. In support ofthis, the control and measurement system may generate a negative drivesignal that causes the driver to resist motion of the conduit when theamplitude of oscillation exceeds the user-controlled value and apositive drive signal that causes the driver to impart motion to theconduit when the amplitude of oscillation is less than theuser-controlled value.

The control and measurement system may include a controller thatgenerates a gain signal based on the sensor signal and a multiplyingdigital-to-analog converter connected to the controller to receive thegain signal and generate the drive signal based on the gain signal.

When the digital flowmeter includes a second sensor connected to theconduit and operable to sense the motion of the conduit, the control andmeasurement system may include a controller that generates themeasurement, a first analog-to-digital converter connected between thefirst sensor and the controller to provide a first digital sensor signalto the controller, and a second analog-to-digital converter connectedbetween the second sensor and the controller to provide a second digitalsensor signal to the controller. The controller may combine the digitalsensor signals to produce a combined signal and to generate a gainsignal based on the first and second digital sensor signals. The controland measurement system also may include a multiplying digital-to-analogconverter connected to receive the combined signal and the gain signalfrom the controller to generate the drive signal as a product of thecombined signal and the gain signal.

The control and measurement system may selectively apply a negative gainto the sensor signal to reduce motion of the conduit.

The control and measurement system also may compensate for zero offsetin the sensor signal. The zero offset may include a componentattributable to gain variation and a component attributable to gainnonlinearity, and the control and measurement system may separatelycompensate for the two components. The control and measurement systemmay compensate for zero offset by generating one or more correctionfactors and modifying the sensor signal using the correction factors.

The control and measurement system may calculate phase offsets for thefirst and second sensor signals. The phase offset may be defined as adifference between a zero-crossing point of a sensor signal and a pointof zero phase for a component of the sensor signal corresponding to afundamental frequency of the sensor signal. The control and measurementsystem may combine the calculated phase offsets to produce a phasedifference.

The control and measurement system may generate the measurement of theproperty by estimating a frequency of the first sensor signal,estimating a frequency of the second sensor signal, with the frequencyof the second sensor signal being different from the frequency of thefirst sensor signal, and calculating a phase difference between thesensor signals using the estimated frequencies.

When the sensor is a velocity sensor, the control and measurement systemmay estimate a frequency, amplitude, and phase of the sensor signal, andmay correct the estimated frequency, amplitude, and phase to account forperformance differences between a velocity sensor and an absoluteposition sensor. Instead of controlling the apparent amplitude ofoscillation (i.e., the velocity of oscillation when the sensor is avelocity sensor), the system may control the true amplitude by dividingthe sensor signal by the estimated frequency. This correction shouldprovide improved amplitude control and noise reduction.

The control and measurement system may estimate a first parameter of thesensor signal, determine a rate of change of a second parameter, andcorrect the estimated first parameter based on the determined rate ofchange. For example, the system may correct an estimated frequency,amplitude, or phase of the sensor signal based on a determined rate ofchange of the frequency or amplitude of oscillation of the conduit. Thesystem may perform separate corrections for each sensor signal.

The digital flowmeter may be a mass flowmeter and the property ofmaterial flowing through the conduit may be a mass flow rate. Thedigital flowmeter also may be a densitometer and the property ofmaterial flowing through the conduit may be a density of the material.

The control and measurement system may account for effects of aerationin the conduit by determining an initial mass flow rate, determining anapparent density of material flowing through the conduit, comparing theapparent density to a known density of the material to determine adensity difference, and adjusting the initial mass flow rate based onthe density difference to produce an adjusted mass flow rate. The systemmay further account for effects of aeration in the conduit by adjustingthe adjusted mass flow rate to account for effects of damping. Tofurther account for effects of aeration in the conduit, the system mayadjust the adjusted mass flow rate based on differences betweenamplitudes of the first and second sensor signals.

The vibratable conduit may include two parallel planar loops. The sensorand driver may be connected between the loops.

The meter may include a power circuit that receives power on only asingle pair of wires. The power circuit provides power to the digitalcontrol and measurement system and to the driver, and the digitalcontrol and measurement system is operable to transmit the measurementof the property of material flowing through the conduit on the singlepair of wires. The power circuit may include a constant output circuitthat provides power to the digital control and measurement system anddrive capacitor that is charged by excess power from the two wires. Thedigital control and measurement system may discharge the drive capacitorto power the driver, and may monitor a charge level of the drivecapacitor and discharge the drive capacitor after a charge level of thecapacitor reaches a threshold level. The digital control and measurementsystem also may discharge the drive capacitor periodically and toperform bidirectional communications on the pair of wires.

The control and measurement system may collect a first data set for aperiod of the periodic signal and process the first data set to generatethe drive signal and the measurement. The system may collect a seconddata set for a subsequent period of the sensor signal simultaneouslywith processing the first data set. The period corresponding to thefirst data set may overlap the period corresponding to the second dataset.

The control and measurement system may control the drive signal tomaintain an amplitude of the sensor signal at a fixed setpoint, reducethe fixed setpoint when the drive signal exceeds a first thresholdlevel, and increase the fixed setpoint when the drive signal is lessthan a second threshold level and the fixed setpoint is less than amaximum permitted value for the setpoint. The first threshold level maybe 95% or less of a maximum permitted drive signal.

The control and measurement system may perform an uncertainty analysison the measurement. In this case, the control and measurement system maytransmit the measurement and results of the uncertainty analysis to thecontrol system.

The control and measurement system may use digital processing to adjusta phase of the drive signal to compensate for a time delay associatedwith the sensor and components connected between the sensor and thedriver.

The details of one or more implementations are set forth in theaccompanying drawings and the description below. Other features andadvantages will be apparent from the description and drawings, and fromthe claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram of a digital mass flowmeter.

FIGS. 2A and 2B are perspective and side views of mechanical componentsof a mass flowmeter.

FIGS. 3A-3C are schematic representations of three modes of motion ofthe flowmeter of FIG. 1.

FIG. 4 is a block diagram of an analog control and measurement circuit.

FIG. 5 is a block diagram of a digital mass flowmeter.

FIG. 6 is a flow chart showing operation of the meter of FIG. 5.

FIGS. 7A and 7B are graphs of sensor data.

FIGS. 8A and 8B are graphs of sensor voltage relative to time.

FIG. 9 is a flow chart of a curve fitting procedure.

FIG. 10 is a flow chart of a procedure for generating phase differences.

FIGS. 11A-11D, 12A-12D, and 13A-13D illustrate drive and sensor voltagesat system startup.

FIG. 14 is a flow chart of a procedure for measuring frequency,amplitude, and phase of sensor data using a synchronous modulationtechnique.

FIGS. 15A and 15B are block diagrams of a mass flowmeter.

FIG. 16 is a flow chart of a procedure implemented by the meter of FIGS.15A and 15B.

FIG. 17 illustrates log-amplitude control of a transfer function.

FIG. 18 is a root locus diagram.

FIGS. 19A-19D are graphs of analog-to-digital converter performancerelative to temperature.

FIGS. 20A-20C are graphs of phase measurements.

FIGS. 21A and 21B are graphs of phase measurements.

FIG. 22 is a flow chart of a zero offset compensation procedure.

FIGS. 23A-23C, 24A, and 24B are graphs of phase measurements.

FIG. 25 is a graph of sensor voltage.

FIG. 26 is a flow chart of a procedure for compensating for dynamiceffects.

FIGS. 27A-35E are graphs illustrating application of the procedure ofFIG. 29.

FIGS. 36A-36L are graphs illustrating phase measurement.

FIG. 37A is a graph of sensor voltages.

FIGS. 37B and 37C are graphs of phase and frequency measurementscorresponding to the sensor voltages of FIG. 37A.

FIGS. 37D and 37E are graphs of correction parameters for the phase andfrequency measurements of FIGS. 37B and 37C.

FIGS. 38A-38H are graphs of raw measurements.

FIGS. 39A-39H are graphs of corrected measurements.

FIGS. 40A-40H are graphs illustrating correction for aeration.

FIG. 41 is a block diagram illustrating the effect of aeration in aconduit.

FIG. 42 is a flow chart of a setpoint control procedure.

FIGS. 43A-43C are graphs illustrating application of the procedure ofFIG. 41.

FIG. 44 is a graph comparing the performance of digital and analogflowmeters.

FIG. 45 is a flow chart showing operation of a self-validating meter.

FIG. 46 is a block diagram of a two-wire digital mass flowmeter.

DETAILED DESCRIPTION

Referring to FIG. 1, a digital mass flowmeter 100 includes a digitalcontroller 105, one or more motion sensors 110, one or more drivers 115,a conduit 120 (also referred to as a flowtube), and a temperature sensor125. The digital controller 105 may be implemented using one or more of,for example, a processor, a field-programmable gate array, an ASIC,other programmable logic or gate arrays, or programmable logic with aprocessor core. The digital controller generates a measurement of massflow through the conduit 120 based at least on signals received from themotion sensors 110. The digital controller also controls the drivers 115to induce motion in the conduit 120. This motion is sensed by the motionsensors 110.

Mass flow through the conduit 120 is related to the motion induced inthe conduit in response to a driving force supplied by the drivers 115.In particular, mass flow is related to the phase and frequency of themotion, as well as to the temperature of the conduit. The digital massflowmeter also may provide a measurement of the density of materialflowing through the conduit. The density is related to the frequency ofthe motion and the temperature of the conduit. Many of the describedtechniques are applicable to a densitometer that provides a measure ofdensity rather than a measure of mass flow.

The temperature in the conduit, which is measured using the temperaturesensor 125, affects certain properties of the conduit, such as itsstiffness and dimensions. The digital controller compensates for thesetemperature effects. The temperature of the digital controller 105affects, for example, the operating frequency of the digital controller.In general, the effects of controller temperature are sufficiently smallto be considered negligible. However, in some instances, the digitalcontroller may measure the controller temperature using a solid statedevice and may compensate for effects of the controller temperature.

A. Mechanical Design

In one implementation, as illustrated in FIGS. 2A and 2B, the conduit120 is designed to be inserted in a pipeline (not shown) having a smallsection removed or reserved to make room for the conduit. The conduit120 includes mounting flanges 12 for connection to the pipeline, and acentral manifold block 16 supporting two parallel planar loops 18 and 20that are oriented perpendicularly to the pipeline. An electromagneticdriver 46 and a sensor 48 are attached between each end of loops 18 and20. Each of the two drivers 46 corresponds to a driver 115 of FIG. 1,while each of the two sensors 48 corresponds to a sensor 120 of FIG. 1.

The drivers 46 on opposite ends of the loops are energized with currentof equal magnitude but opposite sign (i.e., currents that are 180°out-of-phase) to cause straight sections 26 of the loops 18, 20 torotate about their co-planar perpendicular bisector 56, which intersectsthe tube at point P (FIG. 2B). Repeatedly reversing (e.g., controllingsinusoidally) the energizing current supplied to the drivers causes eachstraight section 26 to undergo oscillatory motion that sweeps out a bowtie shape in the horizontal plane about line 56-56, the axis of symmetryof the loop. The entire lateral excursion of the loops at the lowerrounded turns 38 and 40 is small, on the order of 1/16 of an inch for atwo foot long straight section 26 of a pipe having a one inch diameter.The frequency of oscillation is typically about 80 to 90 Hertz.

B. Conduit Motion

The motion of the straight sections of loops 18 and 20 are shown inthree modes in FIGS. 3A, 3B and 3C. In the drive mode shown in FIG. 3B,the loops are driven 180° out-of-phase about their respective points Pso that the two loops rotate synchronously but in the opposite sense.Consequently, respective ends such as A and C periodically come togetherand go apart.

The drive motion shown in FIG. 3B induces the Coriolis mode motion shownin FIG. 3A, which is in opposite directions between the loops and movesthe straight sections 26 slightly toward (or away) from each other. TheCoriolis effect is directly related to mvW, where m is the mass ofmaterial in a cross section of a loop, v is the velocity at which themass is moving (the volumetric flow rate), W is the angular velocity ofthe loop (W=W_(o) sin {dot over (O)}t), and my is the mass flow rate.The Coriolis effect is greatest when the two straight sections aredriven sinusoidally and have a sinusoidally varying angular velocity.Under these conditions, the Coriolis effect is 90° out-of-phase with thedrive signal.

FIG. 3C shows an undesirable common mode motion that deflects the loopsin the same direction. This type of motion might be produced by an axialvibration in the pipeline in the embodiment of FIGS. 2A and 2B becausethe loops are perpendicular to the pipeline.

The type of oscillation shown in FIG. 3B is called the antisymmetricalmode, and the Coriolis mode of FIG. 3A is called the symmetrical mode.The natural frequency of oscillation in the antisymmetrical mode is afunction of the torsional resilience of the legs. Ordinarily theresonant frequency of the antisymmetrical mode for conduits of the shapeshown in FIGS. 2A and 2B is higher than the resonant frequency of thesymmetrical mode. To reduce the noise sensitivity of the mass flowmeasurement, it is desirable to maximize the Coriolis force for a givenmass flow rate. As noted above, the loops are driven at their resonantfrequency, and the Coriolis force is directly related to the frequencyat which the loops are oscillating (i.e., the angular velocity of theloops). Accordingly, the loops are driven in the antisymmetrical mode,which tends to have the higher resonant frequency.

Other implementations may include different conduit designs. Forexample, a single loop or a straight tube section may be employed as theconduit.

C. Electronic Design

The digital controller 105 determines the mass flow rate by processingsignals produced by the sensors 48 (i.e., the motion sensors 110)located at opposite ends of the loops. The signal produced by eachsensor includes a component corresponding to the relative velocity atwhich the loops are driven by a driver positioned next to the sensor anda component corresponding to the relative velocity of the loops due toCoriolis forces induced in the loops. The loops are driven in theantisymmetrical mode, so that the components of the sensor signalscorresponding to drive velocity are equal in magnitude but opposite insign. The resulting Coriolis force is in the symmetrical mode so thatthe components of the sensor signals corresponding to Coriolis velocityare equal in magnitude and sign. Thus, differencing the signals cancelsout the Coriolis velocity components and results in a difference that isproportional to the drive velocity. Similarly, summing the signalscancels out the drive velocity components and results in a sum that isproportional to the Coriolis velocity, which, in turn, is proportionalto the Coriolis force. This sum then may be used to determine the massflow rate.

1. Analog Control System

The digital mass flowmeter 100 provides considerable advantages overtraditional, analog mass flowmeters. For use in later discussion, FIG. 4illustrates an analog control system 400 of a traditional massflowmeter. The sensors 48 each produce a voltage signal, with signalV_(A0) being produced by sensor 48 a and signal V_(B0) being produced bysensor 48 b. V_(A0) and V_(B0) correspond to the velocity of the loopsrelative to each other at the positions of the sensors. Prior toprocessing, signals V_(A0) and V_(B0) are amplified at respective inputamplifiers 405 and 410 to produce signals V_(A1) and V_(B1). To correctfor imbalances in the amplifiers and the sensors, input amplifier 410has a variable gain that is controlled by a balance signal coming from afeedback loop that contains a synchronous demodulator 415 and anintegrator 420.

At the output of amplifier 405, signal V_(A1) is of the form:V _(A1) =V _(D) sin ωt+V _(C) cos ωt,and, at the output of amplifier 410, signal V_(B1) is of the form:V _(B1) =−V _(D) sin ωt+V _(C) cos ωt,where V_(D) and V_(C) are, respectively, the drive voltage and theCoriolis voltage, and ω is the drive mode angular frequency.

Voltages V_(A1) and V_(B1) are differenced by operational amplifier 425to produce:V _(DRV) =V _(A1) −V _(B1)=2V _(D) sin ωt,where V_(DRV) corresponds to the drive motion and is used to power thedrivers. In addition to powering the drivers, V_(DRV) is supplied to apositive going zero crossing detector 430 that produces an output squarewave F_(DRV) having a frequency corresponding to that of V_(DRV)(ω=2πF_(DRV)). F_(DRV) is used as the input to a digital phase lockedloop circuit 435. F_(DRV) also is supplied to a processor 440.

Voltages V_(A1) and V_(B1) are summed by operational amplifier 445 toproduce:V _(COR) =V _(A1) ±V _(B1)2V _(C) cos ωt,where V_(COR) is related to the induced Coriolis motion.

V_(COR) is supplied to a synchronous demodulator 450 that produces anoutput voltage V_(M) that is directly proportional to mass by rejectingthe components of V_(COR) that do not have the same frequency as, andare not in phase with, a gating signal Q. The phase locked loop circuit435 produces Q, which is a quadrature reference signal that has the samefrequency (Ò) as V_(DRV) and is 90° out of phase with V_(DRV) (i.e., inphase with V_(COR)). Accordingly, synchronous demodulator 450 rejectsfrequencies other than Ò so that V_(M) corresponds to the amplitude ofV_(COR) at Ò. This amplitude is directly proportional to the mass in theconduit.

V_(M) is supplied to a voltage-to-frequency converter 455 that producesa square wave signal F_(M) having a frequency that corresponds to theamplitude of V_(M). The processor 440 then divides F_(M) by F_(DRV) toproduce a measurement of the mass flow rate.

Digital phase locked loop circuit 435 also produces a reference signal Ithat is in phase with V_(DRV) and is used to gate the synchronousdemodulator 415 in the feedback loop controlling amplifier 410. When thegains of the input amplifiers 405 and 410 multiplied by the drivecomponents of the corresponding input signals are equal, the summingoperation at operational amplifier 445 produces zero drive component(i.e., no signal in phase with V_(DRV)) in the signal V_(COR). When thegains of the input amplifiers 405 and 410 are not equal, a drivecomponent exists in V_(COR). This drive component is extracted bysynchronous demodulator 415 and integrated by integrator 420 to generatean error voltage that corrects the gain of input amplifier 410. When thegain is too high or too low, the synchronous demodulator 415 produces anoutput voltage that causes the integrator to change the error voltagethat modifies the gain. When the gain reaches the desired value, theoutput of the synchronous modulator goes to zero and the error voltagestops changing to maintain the gain at the desired value.

2. Digital Control System

FIG. 5 provides a block diagram of an implementation 500 of the digitalmass flowmeter 100 that includes the conduit 120, drivers 46, andsensors 48 of FIGS. 2A and 2B, along with a digital controller 505.Analog signals from the sensors 48 are converted to digital signals byanalog-to-digital (“A/D”) converters 510 and supplied to the controller505. The A/D converters may be implemented as separate converters, or asseparate channels of a single converter.

Digital-to-analog (“D/A”) converters 515 convert digital control signalsfrom the controller 505 to analog signals for driving the drivers 46.The use of a separate drive signal for each driver has a number ofadvantages. For example, the system may easily switch betweensymmetrical and antisymmetrical drive modes for diagnostic purposes. Inother implementations, the signals produced by converters 515 may beamplified by amplifiers prior to being supplied to the drivers 46. Instill other implementations, a single D/A converter may be used toproduce a drive signal applied to both drivers, with the drive signalbeing inverted prior to being provided to one of the drivers to drivethe conduit 120 in the antisymmetrical mode.

High precision resistors 520 and amplifiers 525 are used to measure thecurrent supplied to each driver 46. A/D converters 530 convert themeasured current to digital signals and supply the digital signals tocontroller 505. The controller 505 uses the measured currents ingenerating the driving signals.

Temperature sensors 535 and pressure sensors 540 measure, respectively,the temperature and the pressure at the inlet 545 and the outlet 550 ofthe conduit. A/D converters 555 convert the measured values to digitalsignals and supply the digital signals to the controller 505. Thecontroller 505 uses the measured values in a number of ways. Forexample, the difference between the pressure measurements may be used todetermine a back pressure in the conduit. Since the stiffness of theconduit varies with the back pressure, the controller may account forconduit stiffness based on the determined back pressure.

An additional temperature sensor 560 measures the temperature of thecrystal oscillator 565 used by the A/D converters. An A/D converter 570converts this temperature measurement to a digital signal for use by thecontroller 505. The input/output relationship of the A/D convertersvaries with the operating frequency of the converters, and the operatingfrequency varies with the temperature of the crystal oscillator.Accordingly, the controller uses the temperature measurement to adjustthe data provided by the A/D converters, or in system calibration.

In the implementation of FIG. 5, the digital controller 505 processesthe digitized sensor signals produced by the A/D converters 510according to the procedure 600 illustrated in FIG. 6 to generate themass flow measurement and the drive signal supplied to the drivers 46.Initially, the controller collects data from the sensors (step 605).Using this data, the controller determines the frequency of the sensorsignals (step 610), eliminates zero offset from the sensor signals (step615), and determines the amplitude (step 620) and phase (step 625) ofthe sensor signals. The controller uses these calculated values togenerate the drive signal (step 630) and to generate the mass flow andother measurements (step 635). After generating the drive signals andmeasurements, the controller collects a new set of data and repeats theprocedure. The steps of the procedure 600 may be performed serially orin parallel, and may be performed in varying order.

Because of the relationships between frequency, zero offset, amplitude,and phase, an estimate of one may be used in calculating another. Thisleads to repeated calculations to improve accuracy. For example, aninitial frequency determination used in determining the zero offset inthe sensor signals may be revised using offset-eliminated sensorsignals. In addition, where appropriate, values generated for a cyclemay be used as starting estimates for a following cycle.

FIG. 5 provides a general description of the hardware included in adigital flowmeter. A more detailed description of a specific hardwareimplementation is provided in the attached appendix, which isincorporated by reference.

a. Data Collection

For ease of discussion, the digitized signals from the two sensors willbe referred to as signals SV₁ and SV₂, with signal SV₁ coming fromsensor 48 a and signal SV₂ coming from sensor 48 b. Although new data isgenerated constantly, it is assumed that calculations are based upondata corresponding to one complete cycle of both sensors. Withsufficient data buffering, this condition will be true so long as theaverage time to process data is less than the time taken to collect thedata. Tasks to be carried out for a cycle include deciding that thecycle has been completed, calculating the frequency of the cycle (or thefrequencies of SV₁ and SV₂), calculating the amplitudes of SV₁ and SV₂,and calculating the phase difference between SV₁ and SV₂. In someimplementations, these calculations are repeated for each cycle usingthe end point of the previous cycle as the start for the next. In otherimplementations, the cycles overlap by 180° or other amounts (e.g., 90°)so that a cycle is subsumed within the cycles that precede and followit.

FIGS. 7A and 7B illustrate two vectors of sampled data from signals SV₁and SV₂, which are named, respectively, sv1_in and sv2_in. The firstsampling point of each vector is known, and corresponds to a zerocrossing of the sine wave represented by the vector. For sv1_in, thefirst sampling point is the zero crossing from a negative value to apositive value, while for sv2_in the first sampling point is the zerocrossing from a positive value to a negative value.

An actual starting point for a cycle (i.e., the actual zero crossing)will rarely coincide exactly with a sampling point. For this reason, theinitial sampling points (start_sample_SV1 and start_sample_SV2) are thesampling points occurring just before the start of the cycle. To accountfor the difference between the first sampling point and the actual startof the cycle, the approach also uses the position (start_offset_SV1 orstart_offset_SV2) between the starting sample and the next sample atwhich the cycle actually begins.

Since there is a phase offset between signals SV₁ and SV₂, sv1_in andsv2_in may start at different sampling points. If both the sample rateand the phase difference are high, there may be a difference of severalsamples between the start of sv1_in and the start of sv2_in. Thisdifference provides a crude estimate of the phase offset, and may beused as a check on the calculated phase offset, which is discussedbelow. For example, when sampling at 55 kHz, one sample corresponds toapproximately 0.5 degrees of phase shift, and one cycle corresponds toabout 800 sample points.

When the controller employs functions such as the sum (A+B) anddifference (A−B), with B weighted to have the same amplitude as A,additional variables (e.g., start_sample_sum and start_offset_sum) trackthe start of the period for each function. The sum and differencefunctions have a phase offset halfway between SV₁ and SV₂.

In one implementation, the data structure employed to store the datafrom the sensors is a circular list for each sensor, with a capacity ofat least twice the maximum number of samples in a cycle. With this datastructure, processing may be carried out on data for a current cyclewhile interrupts or other techniques are used to add data for afollowing cycle to the lists.

Processing is performed on data corresponding to a full cycle to avoiderrors when employing approaches based on sine-waves. Accordingly, thefirst task in assembling data for a cycle is to determine where thecycle begins and ends. When nonoverlapping cycles are employed, thebeginning of the cycle may be identified as the end of the previouscycle. When overlapping cycles are employed, and the cycles overlap by180°, the beginning of the cycle may be identified as the midpoint ofthe previous cycle, or as the end_point of the cycle preceding theprevious cycle.

The end of the cycle may be first estimated based on the parameters ofthe previous cycle and under the assumption that the parameters will notchange by more than a predetermined amount from cycle to cycle. Forexample, five percent may be used as the maximum permitted change fromthe last cycle's value, which is reasonable since, at sampling rates of55 kHz, repeated increases or decreases of five percent in amplitude orfrequency over consecutive cycles would result in changes of close to5,000 percent in one second.

By designating five percent as the maximum permissible increase inamplitude and frequency, and allowing for a maximum phase change of 5°in consecutive cycles, a conservative estimate for the upper limit onthe end of the cycle for signal SV₁ may be determined as:

${{end\_ sample}{\_ SV1}} \leq {{{start\_ sample}{\_ SV}\; 1} + {\frac{365}{360}*\frac{sample\_ rate}{{est\_ freq}*0.95}}}$

where start_sample_SV1 is the first sample of sv1_in, sample_rate is thesampling rate, and est_freq is the frequency from the previous cycle.The upper limit on the end of the cycle for signal SV₂ (end_sample_SV2)may be determined similarly.

After the end of a cycle is identified, simple checks may be made as towhether the cycle is worth processing. A cycle may not be worthprocessing when, for example, the conduit has stalled or the sensorwaveforms are severely distorted. Processing only suitable cyclesprovides considerable reductions in computation.

One way to determine cycle suitability is to examine certain points of acycle to confirm expected behavior. As noted above, the amplitudes andfrequency of the last cycle give useful starting estimates of thecorresponding values for the current cycle. Using these values, thepoints corresponding to 30°, 150°, 210° and 330° of the cycle may beexamined. If the amplitude and frequency were to match exactly theamplitude and frequency for the previous cycle, these points should havevalues corresponding to est_amp/2, est_amp/2, −est_amp/2, and−est_amp/2, respectively, where est_amp is the estimated amplitude of asignal (i.e., the amplitude from the previous cycle). Allowing for afive percent change in both amplitude and frequency, inequalities may begenerated for each quarter cycle. For the 30° point, the inequality is

${{sv}\; 1{\_ in}( {{{start\_ sample}{\_ SV}\; 1} + {\frac{30}{360}*\frac{sample\_ rate}{{est\_ freq}*1.05}}} )} > {0.475*{est\_ amp}{\_ SV}}$

The inequalities for the other points have the same form, with thedegree offset term (x/360) and the sign of the est_amp_SV1 term havingappropriate values. These inequalities can be used to check that theconduit has vibrated in a reasonable manner.

Measurement processing takes place on the vectors sv1_in(start: end) andsv2 in(start:end) where:start=min(start_sample_(—) SV1,start_sample_(—) SV2), andend=max(end_sample_(—) SV1,end_sample_(—) SV2).

The difference between the start and end points for a signal isindicative of the frequency of the signal.

b. Frequency Determination

The frequency of a discretely-sampled pure sine wave may be calculatedby detecting the transition between periods (i.e., by detecting positiveor negative zero-crossings) and counting the number of samples in eachperiod. Using this method, sampling, for example, an 82.2 Hz sine waveat 55 kHz will provide an estimate of frequency with a maximum error of0.15 percent. Greater accuracy may be achieved by estimating thefractional part of a sample at which the zero-crossing actually occurredusing, for example, start_offset_SV1 and start_offset_SV2. Random noiseand zero offset may reduce the accuracy of this approach.

As illustrated in FIGS. 8A and 8B, a more refined method of frequencydetermination uses quadratic interpolation of the square of the sinewave. With this method, the square of the sine wave is calculated, aquadratic function is fitted to match the minimum point of the squaredsine wave, and the zeros of the quadratic function are used to determinethe frequency. Ifsv _(t) =A sin x _(t)+δ+σε_(t),

where sv_(t) is the sensor voltage at time t, A is the amplitude ofoscillation, x_(t) is the radian angle at time t (i.e., x_(t)=2πft), êis the zero offset, Ë_(t) is a random variable with distribution N(0,1),and Ó is the variance of the noise, then the squared function is givenby:sv _(t) ² =A ² sin² x _(t)±2A(δ+σε_(t))sin x _(t)+2εσε_(t)+δ²+σ²ε_(t) ².

When x_(t) is close to 2π, sin x_(t) and sin² x_(t) can be approximatedas x_(0t)=x_(t)−2ì and x_(0t) ², respectively. Accordingly, for valuesof x_(t) close to 2ì, a_(t) can be approximated as:

$\begin{matrix}{a_{t}^{2} \approx {{A^{2}ϰ_{0t}^{2}} + {2{A( {\delta + {\sigma ɛ}_{t}} )}x_{0t}} + {2{\delta\sigma ɛ}_{t}} + \delta^{2} + {\sigma^{2}ɛ_{t}^{2}}}} \\{\approx {( {{A^{2}ϰ_{0t}^{2}} + {2A\;\delta\; x_{0t}} + \delta^{2}} ) + {{{\sigma ɛ}_{t}( {{2{Ax}_{0t}} + {2\delta} + {{\sigma ɛ}\; t}} )}.}}}\end{matrix}$

This is a pure quadratic (with a non-zero minimum, assuming δ≠0) plusnoise, with the amplitude of the noise being dependent upon both Ó andê. Linear interpolation also could be used.

Error sources associated with this curve fitting technique are randomnoise, zero offset, and deviation from a true quadratic. Curve fittingis highly sensitive to the level of random noise. Zero offset in thesensor voltage increases the amplitude of noise in the sine-squaredfunction, and illustrates the importance of zero offset elimination(discussed below). Moving away from the minimum, the square of even apure sine wave is not entirely quadratic. The most significant extraterm is of fourth order. By contrast, the most significant extra termfor linear interpolation is of third order.

Degrees of freedom associated with this curve fitting technique arerelated to how many, and which, data points are used. The minimum isthree, but more may be used (at greater computational expense) by usingleast-squares fitting. Such a fit is less susceptible to random noise.FIG. 8A illustrates that a quadratic approximation is good up to some20° away from the minimum point. Using data points further away from theminimum will reduce the influence of random noise, but will increase theerrors due to the non-quadratic terms (i.e., fourth order and higher) inthe sine-squared function.

FIG. 9 illustrates a procedure 900 for performing the curve fittingtechnique. As a first step, the controller initializes variables (step905). These variables include end_point, the best estimate of the zerocrossing point; ep_int, the integer value nearest to end_point; s[0 . .. i], the set of all sample points; z[k], the square of the sample pointclosest to end_point; z[0 . . . n−1], a set of squared sample pointsused to calculate end_point; n, the number of sample points used tocalculate end_point (n=2k+1); step length, the number of samples in sbetween consecutive values in z; and iteration_count, a count of theiterations that the controller has performed.

The controller then generates a first estimate of end_point (step 910).The controller generates this estimate by calculating an estimatedzero-crossing point based on the estimated frequency from the previouscycle and searching around the estimated crossing point (forwards andbackwards) to find the nearest true crossing point (i.e., the occurrenceof consecutive samples with different signs). The controller then setsend_point equal to the sample point having the smaller magnitude of thesamples surrounding the true crossing point.

Next, the controller sets n, the number of points for curve fitting(step 915). The controller sets n equal to 5 for a sample rate of 11kHz, and to 21 for a sample rate of 44 kHz. The controller then setsiteration_count to 0 (step 920) and increments iteration_count (step925) to begin the iterative portion of the procedure.

As a first step in the iterative portion of the procedure, thecontroller selects step_length (step 930) based on the value ofiteration count. The controller sets step_length equal to 6, 3, or 1depending on whether iteration_count equals, respectively, 1, 2 or 3.

Next, the controller determines ep_int as the integer portion of the sumof end_point and 0.5 (step 935) and fills the z array (step 940). Forexample, when n equals 5, z[0]=s[ep_int−2*step_length]²,z[1]=s[ep_int−step_length]², z[2]=s[ep_int]²,z[3]=s[ep_int+step_length]², and z[4]=s[ep_int+2*step_length]².

Next, the controller uses a filter, such as a Savitzky-Golay filter, tocalculate smoothed values of z[k−1], z[k] and z[k+1] (step 945).Savitzky-Golay smoothing filters are discussed by Press et al. inNumerical Recipes in C, pp. 650-655 (2nd ed., Cambridge UniversityPress, 1995), which is incorporated by reference. The controller thenfits a quadratic to z[k−1], z[k] and z[k+1] (step 950), and calculatesthe minimum value of the quadratic (z*) and the corresponding position(x*) (step 955).

If x* is between the points corresponding to k−1 and k+1 (step 960),then the controller sets end_point equal to x* (step 965). Thereafter,if iteration_count is less than 3 (step 970), the controller incrementsiteration_count (step 925) and repeats the iterative portion of theprocedure.

If x* is not between the points corresponding to k−1 and k+1 (step 960),or if iteration_count equals 3 (step 970), the controller exits theiterative portion of the procedure. The controller then calculates thefrequency based on the difference between end_point and the startingpoint for the cycle, which is known (step 975).

In essence, the procedure 900 causes the controller to make threeattempts to home in on end_point, using smaller step_lengths in eachattempt. If the resulting minimum for any attempt falls outside of thepoints used to fit the curve (i.e., there has been extrapolation ratherthan interpolation), this indicates that either the previous or newestimate is poor, and that a reduction in step size is unwarranted.

The procedure 900 may be applied to at least three different sine wavesproduced by the sensors. These include signals SV₁ and SV₂ and theweighted sum of the two. Moreover, assuming that zero offset iseliminated, the frequency estimates produced for these signals areindependent. This is clearly true for signals SV₁ and SV₂, as the errorson each are independent. It is also true, however, for the weighted sum,as long as the mass flow and the corresponding phase difference betweensignals SV₁ and SV₂ are large enough for the calculation of frequency tobe based on different samples in each case. When this is true, therandom errors in the frequency estimates also should be independent.

The three independent estimates of frequency can be combined to providean improved estimate. This combined estimate is simply the mean of thethree frequency estimates.

c. Zero Offset Compensation

An important error source in a Coriolis transmitter is zero offset ineach of the sensor voltages. Zero offset is introduced into a sensorvoltage signal by drift in the pre-amplification circuitry and theanalog-to-digital converter. The zero offset effect may be worsened byslight differences in the pre-amplification gains for positive andnegative voltages due to the use of differential circuitry. Each errorsource varies between transmitters, and will vary with transmittertemperature and more generally over time with component wear.

An example of the zero offset compensation technique employed by thecontroller is discussed in detail below. In general, the controller usesthe frequency estimate and an integration technique to determine thezero offset in each of the sensor signals. The controller theneliminates the zero offset from those signals. After eliminating zerooffset from signals SV₁ and SV₂, the controller may recalculate thefrequency of those signals to provide an improved estimate of thefrequency.

d. Amplitude Determination

The amplitude of oscillation has a variety of potential uses. Theseinclude regulating conduit oscillation via feedback, balancingcontributions of sensor voltages when synthesizing driver waveforms,calculating sums and differences for phase measurement, and calculatingan amplitude rate of change for measurement correction purposes.

In one implementation, the controller uses the estimated amplitudes ofsignals SV₁ and SV₂ to calculate the sum and difference of signals SV₁and SV₂, and the product of the sum and difference. Prior to determiningthe sum and difference, the controller compensates one of the signals toaccount for differences between the gains of the two sensors. Forexample, the controller may compensate the data for signal SV₂ based onthe ratio of the amplitude of signal SV₁ to the amplitude of signal SV₂so that both signals have the same amplitude.

The controller may produce an additional estimate of the frequency basedon the calculated sum. This estimate may be averaged with previousfrequency estimates to produce a refined estimate of the frequency ofthe signals, or may replace the previous estimates.

The controller may calculate the amplitude according to a Fourier-basedtechnique to eliminate the effects of higher harmonics. A sensor voltagex(t) over a period T (as identified using zero crossing techniques) canbe represented by an offset and a series of harmonic terms as:x(t)=a ₀/2+a ₁ cos(Òt)+a ₂ cos(2Òt)+a ₃ cos(3Òt)+ . . . +b ₁ sin(Òt)+b ₂sin(2Òt)+ . . .With this representation, a non-zero offset a₀ will result in non-zerocosine terms a_(n). Though the amplitude of interest is the amplitude ofthe fundamental component (i.e., the amplitude at frequency ω),monitoring the amplitudes of higher harmonic components (i.e., atfrequencies kω, where k is greater than 1) may be of value fordiagnostic purposes. The values of a_(n) and b_(n) may be calculated as:

${a_{n} = {\frac{2}{T}{\int_{0}^{T}{{x(t)}\cos\; n\;\omega\ {\mathbb{d}t}}}}},{and}$$b_{n} = {\frac{2}{T}{\int_{0}^{T}{{x(t)}\sin\; n\;\omega\ {{\mathbb{d}t}.}}}}$The amplitude, A_(n), of each harmonic is given by:A _(n)=√{square root over (a _(n) ² +b _(n) ²)}.

The integrals are calculated using Simpson's method with quadraticcorrection (described below). The chief computational expense of themethod is calculating the pure sine and cosine functions.

a. Phase Determination

The controller may use a number of approaches to calculate the phasedifference between signals SV₁ and SV₂. For example, the controller maydetermine the phase offset of each harmonic, relative to the startingtime at t=0, as:

$\varphi_{n} = {\tan^{- 1}{\frac{a_{n}}{b_{n}}.}}$

The phase offset is interpreted in the context of a single waveform asbeing the difference between the start of the cycle (i.e., thezero-crossing point) and the point of zero phase for the component ofSV(t) of frequency ω. Since the phase offset is an average over theentire waveform, it may be used as the phase offset from the midpoint ofthe cycle. Ideally, with no zero offset and constant amplitude ofoscillation, the phase offset should be zero every cycle. The controllermay determine the phase difference by comparing the phase offset of eachsensor voltage over the same time period.

The amplitude and phase may be generated using a Fourier method thateliminates the effects of higher harmonics. This method has theadvantage that it does not assume that both ends of the conduits areoscillating at the same frequency. As a first step in the method, afrequency estimate is produced using the zero crossings to measure thetime between the start and end of the cycle. If linear variation infrequency is assumed, this estimate equals the time-averaged frequencyover the period. Using the estimated, and assumed time-invariant,frequency ω of the cycle, the controller calculates:

${I_{1} = {\frac{2\;\omega}{\pi}{\int_{0}^{- \frac{2\;\pi}{\omega}}{{{SV}(t)}{\sin( {\omega\; t} )}\ {\mathbb{d}t}}}}},{{{and}\mspace{14mu} I_{2}} = {\frac{2\;\omega}{\pi}{\int_{0}^{- \frac{2\;\pi}{\omega}}{{{SV}(t)}{\cos( {\omega\; t} )}\ {\mathbb{d}t}}}}},$

where SV(t) is the sensor voltage waveform (i.e., SV₁(t) or SV₂(t)). Thecontroller then determines the estimates of the amplitude and phase:

${{Amp} = \sqrt{I_{1}^{2} + I_{2}^{2}}},{{{and}\mspace{14mu}{Phase}} = {\tan^{- 1}{\frac{I_{2}}{I_{1}}.}}}$

The controller then calculates a phase difference, assuming that theaverage phase and frequency of each sensor signal is representative ofthe entire waveform. Since these frequencies are different for SV₁ andSV₂, the corresponding phases are scaled to the average frequency. Inaddition, the phases are shifted to the same starting point (i.e., themidpoint of the cycle on SV₁). After scaling, they are subtracted toprovide the phase difference:

$\mspace{79mu}{{{{scaled\_ phase}{\_{SV}}_{1}} = {{{phase\_}{SV}}_{1}\frac{av\_ freq}{{{freq\_}{SV}}_{1}}}},{{{scaled\_ shift}{\_{SV}}_{2}} = \frac{( {{{midpoint\_}{SV}}_{2} - {{midpoint\_}{SV}}_{1}} )h\;{{freq\_}{SV}}_{2}}{360}},\mspace{79mu}{and}}$${{{scaled\_ phase}{\_{SV}}_{2}} = {( {{{phase\_}{SV}}_{2} + {{scale\_ shift}{\_{SV}}_{2}}} )\frac{av\_ freq}{{{freq\_}{SV}}_{2}}}},$

where h is the sample length and the midpoints are defined in terms ofsamples:

${{midpoint\_}{SV}}_{x} = \frac{( {{{startpoint\_}{SV}}_{x} + {{endpoint\_}{SV}}_{x}} )}{2}$

In general, phase and amplitude are not calculated over the sametime-frame for the two sensors. When the flow rate is zero, the twocycle mid-points are coincident. However, they diverge at high flowrates so that the calculations are based on sample sets that are notcoincident in time. This leads to increased phase noise in conditions ofchanging mass flow. At full flow rate, a phase shift of 4° (out of 360°)means that only 99% of the samples in the SV₁ and SV₂ data sets arecoincident. Far greater phase shifts may be observed under aeratedconditions, which may lead to even lower rates of overlap.

FIG. 10 illustrates a modified approach 1000 that addresses this issue.First, the controller finds the frequencies (f₁, f₂) and the mid-points(m₁, m₂) of the SV₁ and SV₂ data sets (d₁, d₂) (step 1005). Assuminglinear shift in frequency from the last cycle, the controller calculatesthe frequency of SV₂ at the midpoint of SV₁ (f_(2m1)) and the frequencyof SV₁ at the midpoint of SV₂ (f_(1m2)) (step 1010).

The controller then calculates the starting and ending points of newdata sets (d_(1m2) and d_(2m1)) with mid-points m₂ and m₁ respectively,and assuming frequencies of f_(1m2) and f_(2ml)(step 1015). These endpoints do not necessarily coincide with zero crossing points. However,this is not a requirement for Fourier-based calculations.

The controller then carries out the Fourier calculations of phase andamplitude on the sets d₁ and d_(2m1), and the phase differencecalculations outlined above (step 1020). Since the mid-points of d₁ andd_(2m1) are identical, scale-shift S V₂ is always zero and can beignored. The controller repeats these calculations for the data sets d₂and d_(1m2) (step 1025). The controller then generates averages of thecalculated amplitude and phase difference for use in measurementgeneration (step 1030). When there is sufficient separation between themid points m₁ and m₂, the controller also may use the two sets ofresults to provide local estimates of the rates of change of phase andamplitude.

The controller also may use a difference-amplitude method that involvescalculating a difference between SV₁ and SV₂, squaring the calculateddifference, and integrating the result. According to another approach,the controller synthesizes a sine wave, multiplies the sine wave by thedifference between signals SV₁ and SV₂, and integrates the result. Thecontroller also may integrate the product of signals SV₁ and SV₂, whichis a sine wave having a frequency 2f (where f is the average frequencyof signals SV₁ and SV₂), or may square the product and integrate theresult. The controller also may synthesize a cosine wave comparable tothe product sine wave and multiply the synthesized cosine wave by theproduct sine wave to produce a sine wave of frequency 4f that thecontroller then integrates. The controller also may use multiple ones ofthese approaches to produce separate phase measurements, and then maycalculate a mean value of the separate measurements as the final phasemeasurement.

The difference-amplitude method starts with:

${{{SV}_{1}(t)} = {{A_{1}{\sin( {{2\;\pi\;{ft}} + \frac{\varphi}{2}} )}\mspace{14mu}{and}\mspace{14mu}{{SV}_{2}(t)}} = {A_{2}{\sin( {{2\;\pi\;{ft}} - \frac{\varphi}{2}} )}}}},$

where ô is the phase difference between the sensors. Basic trigonometricidentities may be used to define the sum (Sum) and difference (Diff)between the signals as:

${{{Sum} \equiv {{{SV}_{1}(t)} + {\frac{A_{1}}{A_{2}}{{SV}_{2}(t)}}}} = {2\; A_{1}\cos{\frac{\varphi}{2} \cdot \sin}\; 2\;\pi\;{ft}}},{and}$${{Diff} \equiv {{{SV}_{1}(t)} - {\frac{A_{1}}{A_{2}}{{SV}_{2}(t)}}}} = {2\; A_{1}\sin{\frac{\varphi}{2} \cdot \cos}\; 2\;\pi\;{{ft}.}}$

These functions have amplitudes of 2A₁ cos(ô2) and 2A₁ sin(ô2),respectively. The controller calculates data sets for Sum and Diff fromthe data for SV₁ and SV₂, and then uses one or more of the methodsdescribed above to calculate the amplitude of the signals represented bythose data sets. The controller then uses the calculated amplitudes tocalculate the phase difference, ô.

As an alternative, the phase difference may be calculated using thefunction Prod, defined as:

$\begin{matrix}{{{Prod} \equiv {SumxDiff}} = {4\; A_{1}^{2}\cos{\frac{\varphi}{2} \cdot \sin}{\frac{\varphi}{2} \cdot \cos}\; 2\;\pi\;{{ft} \cdot \sin}\; 2\;\pi\;{ft}}} \\{{= {A_{1}^{2}\sin\;{\varphi \cdot \sin}\; 4\;\pi\;{ft}}},}\end{matrix}$

which is a function with amplitude A² sin ô and frequency 2f. Prod canbe generated sample by sample, and ô may be calculated from theamplitude of the resulting sine wave.

The calculation of phase is particularly dependent upon the accuracy ofprevious calculations (i.e., the calculation of the frequencies andamplitudes of SV₁ and SV₂). The controller may use multiple methods toprovide separate (if not entirely independent) estimates of the phase,which may be combined to give an improved estimate.

a. Drive Signal Generation

The controller generates the drive signal by applying a gain to thedifference between signals SV₁ and SV₂. The controller may apply eithera positive gain (resulting in positive feedback) or a negative gain(resulting in negative feedback).

In general, the Q of the conduit is high enough that the conduit willresonate only at certain discrete frequencies. For example, the lowestresonant frequency for some conduits is between 65 Hz and 95 Hz,depending on the density of the process fluid, and irrespective of thedrive frequency. As such, it is desirable to drive the conduit at theresonant frequency to minimize cycle-to-cycle energy loss. Feeding backthe sensor voltage to the drivers permits the drive frequency to migrateto the resonant frequency.

As an alternative to using feedback to generate the drive signal, puresine waves having phases and frequencies determined as described abovemay be synthesized and sent to the drivers. This approach offers theadvantage of eliminating undesirable high frequency components, such asharmonics of the resonant frequency. This approach also permitscompensation for time delays introduced by the analog-to-digitalconverters, processing, and digital-to-analog converters to ensure thatthe phase of the drive signal corresponds to the mid-point of the phasesof the sensor signals. This compensation may be provided by determiningthe time delay of the system components and introducing a phase shiftcorresponding to the time delay.

Another approach to driving the conduit is to use square wave pulses.This is another synthesis method, with fixed (positive and negative)direct current sources being switched on and off at timed intervals toprovide the required energy. The switching is synchronized with thesensor voltage phase. Advantageously, this approach does not requiredigital-to-analog converters.

In general, the amplitude of vibration of the conduit should rapidlyachieve a desired value at startup, so as to quickly provide themeasurement function, but should do so without significant overshoot,which may damage the meter. The desired rapid startup may be achieved bysetting a very high gain so that the presence of random noise and thehigh Q of the conduit are sufficient to initiate motion of the conduit.In one implementation, high gain and positive feedback are used toinitiate motion of the conduit. Once stable operation is attained, thesystem switches to a synthesis approach for generating the drivesignals.

Referring to FIGS. 11A-13D, synthesis methods also may be used toinitiate conduit motion when high gain is unable to do so. For example,if the DC voltage offset of the sensor voltages is significantly largerthan random noise, the application of a high gain will not induceoscillatory motion. This condition is shown in FIGS. 11A-11D, in which ahigh gain is applied at approximately 0.3 seconds. As shown in FIGS. 11Aand 11B, application of the high gain causes one of the drive signals toassume a large positive value (FIG. 11A) and the other to assume a largenegative value (FIG. 11B). The magnitudes of the drive signals vary withnoise in the sensor signals (FIGS. 11C and 11D). However, the amplifiednoise is insufficient to vary the sign of the drive signals so as toinduce oscillation.

FIGS. 12A-12D illustrate that imposition of a square wave over severalcycles can reliably cause a rapid startup of oscillation. Oscillation ofa conduit having a two inch diameter may be established in approximatelytwo seconds. The establishment of conduit oscillation is indicated bythe reduction in the amplitude of the drive signals, as shown in FIGS.12A and 12B. FIGS. 13A-13D illustrate that oscillation of a one inchconduit may be established in approximately half a second.

A square wave also may be used during operation to correct conduitoscillation problems. For example, in some circumstances, flow meterconduits have been known to begin oscillating at harmonics of theresonant frequency of the conduit, such as frequencies on the order of1.5 kHz. When such high frequency oscillations are detected, a squarewave having a more desirable frequency may be used to return the conduitoscillation to the resonant frequency.

a. Measurement Generation

The controller digitally generates the mass flow measurement in a mannersimilar to the approach used by the analog controller. The controlleralso may generate other measurements, such as density.

In one implementation, the controller calculates the mass flow based onthe phase difference in degrees between the two sensor signals(phase_diff), the frequency of oscillation of the conduit (freq), andthe process temperature (temp):T _(z)=temp−T _(c),noneu_(—) mf=tan(i*phase_diff/180), andmassflow=16(MF ₁ *T _(z) ² +MF ₂ *T _(z) +MF ₃)*noneu_(—) mf/freq,

where T_(c) is a calibration temperature, MF₁-MF₃ are calibrationconstants calculated during a calibration procedure, and noneu_mf is themass flow in non-engineering units.

The controller calculates the density based on the frequency ofoscillation of the conduit and the process temperature:T _(z)=temp−T _(c),c ₂=freq², anddensity=(D ₁ *T _(z) ² +D ₂ *T _(z) +D ₃)/c ₂ +D ₄ *T _(z) ²,

where D₁-D₄ are calibration constants generated during a calibrationprocedure.

a. D. Integration Techniques

Many integration techniques are available, with different techniquesrequiring different levels of computational effort and providingdifferent levels of accuracy. In the described implementation, variantsof Simpson's method are used. The basic technique may be expressed as:

${{\int_{t_{n}}^{t_{n + 2}}{y\ {\mathbb{d}t}}} \approx {\frac{h}{3}( {y_{n} + {4\; y_{n + 1}} + y_{n + 2}} )}},$

where t_(k) is the time at sample k, y_(k) is the corresponding functionvalue, and h is the step length. This rule can be applied repeatedly toany data vector with an odd number of data points (i.e., three or morepoints), and is equivalent to fitting and integrating a cubic spline tothe data points. If the number of data points happens to be even, thenthe so-called ⅜ths rule can be applied at one end of the interval:

${\int_{t_{n}}^{t_{n + 3}}{y\ {\mathbb{d}t}}} \approx {\frac{3h}{8}{( {y_{n} + {3y_{n + 1}} + {3y_{n + 2}} + y_{n + 3}} ).}}$

As stated earlier, each cycle begins and ends at some offset (e.g.,start_offset_SV1) from a sampling point. The accuracy of the integrationtechniques are improved considerably by taking these offsets intoaccount. For example, in an integration of a half cycle sine wave, theareas corresponding to partial samples must be included in thecalculations to avoid a consistent underestimate in the result.

Two types of function are integrated in the described calculations:either sine or sine-squared functions. Both are easily approximatedclose to zero where the end points occur. At the end points, the sinewave is approximately linear and the sine-squared function isapproximately quadratic.

In view of these two types of functions, three different integrationmethods have been evaluated. These are Simpson's method with no endcorrection, Simpson's method with linear end correction, and Simpson'smethod with quadratic correction.

The integration methods were tested by generating and sampling pure sineand sine-squared functions, without simulating any analog-to-digitaltruncation error. Integrals were calculated and the results werecompared to the true amplitudes of the signals. The only source of errorin these calculations was due to the integration techniques. The resultsobtained are illustrated in tables A and B.

TABLE A Integration of a sine function % error (based on 1000 Av. ErrorsS.D. Error Max. Error simulations) (%) (%) (%) Simpson Only −3.73e−3 1.33e−3 6.17e−3 Simpson + linear correction 3.16e−8 4.89e−8 1.56e−7Simpson + quadratic correction 2.00e−4 2.19e−2 5.18e−1

TABLE B Integration of a sine-squared function % error (based on 1000Av. Errors S.D. Error Max. Error simulations (%) (%) (%) Simpson Only2.21e−6 1.10e−6 4.39e−3 Simpson + linear correction 2.21e−6 6.93e−72.52e−6 Simpson + quadratic correction  2.15e−11  6.83e−11  1.88e−10

For sine functions, Simpson's method with linear correction was unbiasedwith the smallest standard deviation, while Simpson's method withoutcorrection was biased to a negative error and Simpson's method withquadratic correction had a relatively high standard deviation. Forsine-squared functions, the errors were generally reduced, with thequadratic correction providing the best result. Based on theseevaluations, linear correction is used when integrating sine functionsand quadratic correction is used when integrating sine-squaredfunctions.

E. Synchronous Modulation Technique

FIG. 14 illustrates an alternative procedure 1400 for processing thesensor signals. Procedure 1400 is based on synchronous modulation, suchas is described by Denys et al., in “Measurement of Voltage Phase forthe French Future Defence Plan Against Losses of Synchronism”, IEEETransactions on Power Delivery, 7(1), 62-69, 1992 and by Begovic et al.in “Frequency Tracking in Power Networks in the Presence of Harmonics”,IEEE Transactions on Power Delivery, 8(2), 480-486, 1993, both of whichare incorporated by reference.

First, the controller generates an initial estimate of the nominaloperating frequency of the system (step 1405). The controller thenattempts to measure the deviation of the frequency of a signal x[k](e.g., SV₁) from this nominal frequency:x[k]=A sin [(Ò ₀ +,Ò)kh+â]+Ë(k),

where A is the amplitude of the sine wave portion of the signal, Ò₀ isthe nominal frequency (e.g., 88 Hz), Ò is the deviation from the nominalfrequency, h is the sampling interval, â is the phase shift, and Ë(k)corresponds to the added noise and harmonics.

To generate this measurement, the controller synthesizes two signalsthat oscillate at the nominal frequency (step 1410). The signals arephase shifted by 0 and ì/2 and have amplitude of unity. The controllermultiplies each of these signals by the original signal to producesignals y₁ and y₂ (step 1415):

${y_{1} = {{{x\lbrack k\rbrack}{\cos( {\omega_{0}{kh}} )}} = {{\frac{A}{2}{\sin\lbrack {{( {{2\omega_{0}} + {\Delta\;\omega}} ){kh}} + \Phi} \rbrack}} + {\frac{A}{2}{\sin( {{\Delta\;\omega\;{kh}} + \Phi} )}}}}},{and}$${y_{2} = {{{x\lbrack k\rbrack}{\sin( {\omega_{0}\;{kh}} )}} = {{\frac{A}{2}{\cos\lbrack {{( {{2\omega_{0}} + {\Delta\;\omega}} ){kh}} + \Phi} \rbrack}} + {\frac{A}{2}{\cos( {{\Delta\;\omega\;{kh}} + \Phi} )}}}}},$

where the first terms of y₁ and y₂ are high frequency (e.g., 176 Hz)components and the second terms are low frequency (e.g., 0 Hz)components. The controller then eliminates the high frequency componentsusing a low pass filter (step 1420):

${y_{1}^{/} = {{\frac{A}{2}{\sin( {{\Delta\;\omega\;{kh}} + \Phi} )}} + {ɛ_{1}\lbrack k\rbrack}}},{and}$${y_{1}^{/} = {{\frac{A}{2}{\cos( {{\Delta\;\omega\;{kh}} + \Phi} )}} + {ɛ_{2}\lbrack k\rbrack}}},$

where ε₁[k] and ε₂[k] represent the filtered noise from the originalsignals. The controller combines these signals to produce u[k] (step1425):

$\begin{matrix}{{u\lbrack k\rbrack} = {( {{y_{1}^{/}\lbrack k\rbrack} + {j\;{y_{2}^{/}\lbrack k\rbrack}}} )( {{y_{1}^{/}\lbrack {k - 1} \rbrack} + {j\;{y_{2}^{/}\lbrack {k - 1} \rbrack}}} )}} \\{= {{u_{1}\lbrack k\rbrack} + {j\;{u_{2}\lbrack k\rbrack}}}} \\{{= {{\frac{A^{2}}{4}{\cos( {{\Delta\omega}\; h} )}} + {j\frac{A^{2}}{4}{\sin( {\Delta\;\omega\; h} )}}}},}\end{matrix}$

which carries the essential information about the frequency deviation.As shown, u₁[k] represents the real component of u[k], while u₂[k]represents the imaginary component.

The controller uses the real and imaginary components of u[k] tocalculate the frequency deviation, ,f (step 1430):

${\Delta\; f} = {\frac{1}{h}\arctan{\frac{u_{2}\lbrack k\rbrack}{u_{1}\lbrack k\rbrack}.}}$

The controller then adds the frequency deviation to the nominalfrequency (step 1435) to give the actual frequency:f=Δf+f ₀.

The controller also uses the real and imaginary components of u[k] todetermine the amplitude of the original signal. In particular, thecontroller determines the amplitude as (step 1440):A ²=4√{square root over (u ₁ ² [k]+u ₂ ² [k])}.

Next, the controller determines the phase difference between the twosensor signals (step 1445). Assuming that any noise (ε₁[k] and ε₂[k])remaining after application of the low pass filter described below willbe negligible, noise free versions of y₁′[k] and y₂′[k] (y₁*[k] andy₂*[k]) may be expressed as:

${{y_{1}^{*}\lbrack k\rbrack} = {\frac{A}{2}{\sin( {{\Delta\;\omega\;{kh}} + \Phi} )}}},{and}$${y_{2}^{*}\lbrack k\rbrack} = {\frac{A}{2}{{\cos( {{\Delta\;\omega\;{kh}} - \Phi} )}.}}$

Multiplying these signals together gives:

$v = {{y_{1}^{*}y_{2}^{*}} = {{\frac{A^{2}}{8}\lbrack {{\sin( {2\Phi} )} + {\sin( {2\Delta\;\omega\;{kh}} )}} \rbrack}.}}$

Filtering this signal by a low pass filter having a cutoff frequencynear 0 Hz removes the unwanted component and leaves:

${v^{\prime} = {\frac{A^{2}}{8}{\sin( {2\Phi} )}}},$

from which the phase difference can be calculated as:

$\Phi = {\frac{1}{2}\arcsin{\frac{8v^{\prime}}{A^{2}}.}}$

This procedure relies on the accuracy with which the operating frequencyis initially estimated, as the procedure measures only the deviationfrom this frequency. If a good estimate is given, a very narrow filtercan be used, which makes the procedure very accurate. For typicalflowmeters, the operating frequencies are around 95 Hz (empty) and 82 Hz(full). A first approximation of half range (88 Hz) is used, whichallows a low-pass filter cut-off of 13 Hz. Care must be taken inselecting the cut-off frequency as a very small cut-off frequency canattenuate the amplitude of the sine wave.

The accuracy of measurement also depends on the filteringcharacteristics employed. The attenuation of the filter in the dead-banddetermines the amount of harmonics rejection, while a smaller cutofffrequency improves the noise rejection.

F. Meter with PI Control

FIGS. 15A and 15B illustrate a meter 1500 having a controller 1505 thatuses another technique to generate the signals supplied to the drivers.Analog-to-digital converters 1510 digitize signals from the sensors 48and provide the digitized signals to the controller 1505. The controller1505 uses the digitized signals to calculate gains for each driver, withthe gains being suitable for generating desired oscillations in theconduit. The gains may be either positive or negative. The controller1505 then supplies the gains to multiplying digital-to-analog converters1515. In other implementations, two or more multiplyingdigital-to-analog converters arranged in series may be used to implementa single, more sensitive multiplying digital-to-analog converter.

The controller 1505 also generates drive signals using the digitizedsensor signals. The controller 1505 provides these drive signals todigital-to-analog converters 1520 that convert the signals to analogsignals that are supplied to the multiplying digital-to-analogconverters 1515.

The multiplying digital-to-analog converters 1515 multiply the analogsignals by the gains from the controller 1505 to produce signals fordriving the conduit. Amplifiers 1525 then amplify these signals andsupply them to the drivers 46. Similar results could be obtained byhaving the controller 1505 perform the multiplication performed by themultiplying digital-to-analog converter, at which point the multiplyingdigital-to-analog converter could be replaced by a standarddigital-to-analog converter.

FIG. 15B illustrates the control approach in more detail. Within thecontroller 1505, the digitized sensor signals are provided to anamplitude detector 1550, which determines a measure, a(t), of theamplitude of motion of the conduit using, for example, the techniquedescribed above. A summer 1555 then uses the amplitude a(t) and adesired amplitude a₀ to calculate an error e(t) as:e(t)=a ₀ −a(t).

The error e(t) is used by a proportional-integral (“PI”) control block1560 to generate a gain K₀(t). This gain is multiplied by the differenceof the sensor signals to generate the drive signal. The PI control blockpermits high speed response to changing conditions. The amplitudedetector 1550, summer 1555, and PI control block 1560 may be implementedas software processed by the controller 1505, or as separate circuitry.

1. Control Procedure

The meter 1500 operates according to the procedure 1600 illustrated inFIG. 16. Initially, the controller receives digitized data from thesensors (step 1605). Thereafter, the procedure 1600 includes threeparallel branches: a measurement branch 1610, a drive signal generationbranch 1615, and a gain generation branch 1620.

In the measurement branch 1610, the digitized sensor data is used togenerate measurements of amplitude, frequency, and phase, as describedabove (step 1625). These measurements then are used to calculate themass flow rate (step 1630) and other process variables. In general, thecontroller 1505 implements the measurement branch 1610.

In the drive signal generation branch 1615, the digitized signals fromthe two sensors are differenced to generate the signal (step 1635) thatis multiplied by the gain to produce the drive signal. As describedabove, this differencing operation is performed by the controller 1505.In general, the differencing operation produces a weighted differencethat accounts for amplitude differences between the sensor signals.

In the gain generation branch 1620, the gain is calculated using theproportional-integral control block. As noted above, the amplitude,a(t), of motion of the conduit is determined (step 1640) and subtractedfrom the desired amplitude a₀ (step 1645) to calculate the error e(t).Though illustrated as a separate step, generation of the amplitude,a(t), may correspond to generation of the amplitude in the measurementgeneration step 1625. Finally, the PI control block uses the error e(t)to calculate the gain (step 1650).

The calculated gain is multiplied by the difference signal to generatethe drive signal supplied to the drivers (step 1655). As describedabove, this multiplication operation is performed by the multiplying D/Aconverter or may be performed by the controller.

2. PI Control Block

The objective of the PI control block is to sustain in the conduit puresinusoidal oscillations having an amplitude a₀. The behavior of theconduit may be modeled as a simple mass-spring system that may beexpressed as:{umlaut over (x)}+2ζω_(n) {dot over (x)}+ω _(n) ² x=0,

where x is a function of time and the displacement of the mass fromequilibrium, Ò_(n) is the natural frequency, and ë is a damping factor,which is assumed to be small (e.g., 0.001). The solution to this forceequation as a function of an output y(t) and an input i(t) is analogousto an electrical network in which the transfer function between asupplied current, i(s), and a sensed output voltage, y(s), is:

$\frac{y(s)}{i(s)} = {\frac{ks}{s^{2} + {2{\zeta\omega}_{n}s} + \omega_{n}^{2}}.}$

To achieve the desired oscillation in the conduit, a positive-feedbackloop having the gain K₀(t) is automatically adjusted by a ‘slow’ outerloop to give:{umlaut over (x)}+(2ζω_(n) −kK ₀(t)){dot over (x)}+ω _(n) ² x=0.

The system is assumed to have a “two-time-scales” property, which meansthat variations in K₀(t) are slow enough that solutions to the equationfor x provided above can be obtained by assuming constant damping.

A two-term PI control block that gives zero steady-state error may beexpressed as:K ₀(t)=K _(p) e(t)+K _(i)∫₀ ^(t) e(t)dt,where the error, e(t) (i.e., a₀−a(t)), is the input to the PI controlblock, and K_(p) and K_(i) are constants. In one implementation, witha₀=10, controller constants of K_(p)=0.02 and K_(i)=0.0005 provide aresponse in which oscillations build up quickly. However, this PIcontrol block is nonlinear, which may result in design and operationaldifficulties.

A linear model of the behavior of the oscillation amplitude may bederived by assuming that x(t) equals AË^(jÒt), which results in:{dot over (x)}={dot over (A)}e ^(jωt) +jωe ^(jωt), and{umlaut over (x)}[Ä−ω ² A]e ^(jωt)+2jω{dot over (A)}e ^(jωt).

Substituting these expressions into the expression for oscillation ofthe loop, and separating into real and imaginary terms, gives:jω{2{dot over (A)}+(2ζω_(n) −kK ₀)A}=0, andÄ+(2ζω_(n) −kK ₀){dot over (A)}+(ω_(n) ²−ω²)A=0.

A(t) also may be expressed as:

$\frac{\overset{.}{A}}{A} = {{{- \zeta}\;\omega_{n}} + {\frac{k\; K_{0}}{2}{t.}}}$

A solution of this equation is:

${\log\;{A(t)}} = {( {{{- \zeta}\;\omega_{n}} + \frac{k\; K_{0}}{2}} ){t.}}$

Transforming variables by defining a(t) as being equal to log A(t), theequation for A(t) can be written as:

${\frac{\mathbb{d}a}{\mathbb{d}t} = {{{- \zeta}\;\omega_{n}} + \frac{k\;{K_{0}(t)}}{2}}},$

where K_(o) is now explicitly dependent on time. Taking Laplacetransforms results in:

${{a(s)} = \frac{{{- \zeta}\;\omega_{n}} - {k\;{{K_{0}(s)}/2}}}{s}},$

which can be interpreted in terms of transfer-functions as in FIG. 17.This figure is of particular significance for the design of controllers,as it is linear for all K_(o) and a, with the only assumption being thetwo-time-scales property. The performance of the closed-loop is robustwith respect to this assumption so that fast responses which areattainable in practice can be readily designed.

From FIG. 17, the term ëÒ_(n), is a “load disturbance” that needs to beeliminated by the controller (i.e., kK_(o)/2 must equal ëÒ_(n) for a(t)to be constant). For zero steady-state error this implies that theouter-loop controller must have an integrator (or very large gain). Assuch, an appropriate PI controller, C(s), may be assumed to beK_(p)(1+1/sT_(i)), where T_(i) is a constant. The proportional term isrequired for stability. The term ëÒ_(n), however, does not affectstability or controller design, which is based instead on the open-looptransfer function:

${{C(s)}{G(s)}} = {\frac{a(s)}{e(s)} = {\frac{k\;{K_{p}( {1 + {s\; T_{i\;}}} )}}{2s^{2}T_{i}} = {\frac{k\;{K_{p}/2}( {s + {1/\; T_{i\;}}} )}{s^{2}}.}}}$

The root locus for varying K_(p) is shown in FIG. 18. For small K_(p),there are slow underdamped roots. As K_(p) increases, the roots becomereal at the point P for which the controller gain is K_(p)=8/(kT_(i)).Note in particular that the theory does not place any restriction uponthe choice of T_(i). Hence the response can, in principle, be madecritically damped and as fast as desired by appropriate choices of K_(p)and T_(i).

Although the poles are purely real at point P, this does not mean thereis no overshoot in the closed-loop step response. This is most easilyseen by inspecting the transfer function between the desired value, a₀,and the error e:

${\frac{e(s)}{a_{0}(s)} = {\frac{s^{2}}{s^{2} + {0.5{{kK}_{p}( {s + {1/T_{i}}} )}}} = \frac{s^{2}}{p_{2}(s)}}},$

where p₂ is a second-order polynomial. With a step input, a_(o)(s)=É/s,the response can be written as Ép′(t), where p(t) is the inversetransform of 1/p₂(s) and equals a₁ exp(−Î₁t)+a₂ exp(−Î₂t). The signalp(t) increases and then decays to zero so that e(t), which isproportional to p′, must change sign, implying overshoot in a(t). Theset-point a_(o) may be prefiltered to give a pseudo set-point a_(o)*:

${{a_{0}^{*}(s)} = {\frac{1}{1 + {sT}_{i}}a_{0}(s)}},$

where T_(i) is the known controller parameter. With this prefilter, realcontroller poles should provide overshoot-free step responses. Thisfeature is useful as there may be physical constraints on overshoot(e.g., mechanical interference or overstressing of components).

The root locus of FIG. 18 assumes that the only dynamics are from theinner-loop's gain/log-amplitude transfer function (FIG. 16) and theouter-loop's PI controller C(s) (i.e., that the log-amplitude a=log A ismeasured instantaneously). However, A is the amplitude of an oscillationwhich might be growing or decaying and hence cannot in general bemeasured without taking into account the underlying sinusoid. There areseveral possible methods for measuring A, in addition to those discussedabove. Some are more suitable for use in quasi-steady conditions. Forexample, a phase-locked loop in which a sinusoidal signals(t)=sin(Ò_(n)t+â₀) locks onto the measured waveformy(t)=A(t)sin(Ò_(n)t+â₁) may be employed. Thus, a measure of theamplitude a=log A is given by dividing these signals (with appropriatesafeguards and filters). This method is perhaps satisfactory near thesteady-state but not for start-up conditions before there is a lock.

Another approach uses a peak-follower that includes a zero-crossingdetector together with a peak-following algorithm implemented in thecontroller. Zero-crossing methods, however, can be susceptible to noise.In addition, results from a peak-follower are available only everyhalf-cycle and thereby dictate the sample interval for controllerupdates.

Finally, an AM detector may be employed. Given a sine wave y(t)=A sinÒ_(n)t, an estimate of A may be obtained from Â₁0.5ìF{abs(y)}, where F{} is a suitable low-pass filter with unity DC gain. The AM detector isthe simplest approach. Moreover, it does not presume that there areoscillations of any particular frequency, and hence is usable duringstartup conditions. It suffers from a disadvantage that there is aleakage of harmonics into the inner loop which will affect the spectrumof the resultant oscillations. In addition, the filter adds extradynamics into the outer loop such that compromises need to be madebetween speed of response and spectral purity. In particular, an effectof the filter is to constrain the choice of the best T_(i).

The Fourier series for abs(y) is known to be:

${A\mspace{14mu}{{abs}( {\sin\mspace{14mu}\omega_{n}t} )}} = {{\frac{2A}{\pi}\lbrack {1 + {\frac{2}{3}\cos\; 2\omega_{n}t} - {\frac{2}{15}\cos\; 4\;\omega_{n}t} + {\frac{2}{35}\cos\; 6\omega_{n}t} + \ldots}\mspace{14mu} \rbrack}.}$

As such, the output has to be scaled by π/2 to give the correct DCoutput A, and the (even) harmonic terms a_(k) cos 2kω_(n)t have to befiltered out. As all the filter needs to do is to pass the DC componentthrough and reduce all other frequencies, a “brick-wall” filter with acut-off below 2ω_(n) is sufficient. However, the dynamics of the filterwill affect the behavior of the closed-loop. A common choice of filteris in the Butterworth form. For example, the third-order low-pass filterwith a design break-point frequency ω_(b) is:

${F(s)} = {\frac{1}{1 + {2{s/\omega_{b}}} + {2{s^{2}/\omega_{b}^{2}}} + {s^{3}/\omega_{b}^{3}}}.}$

At the design frequency the response is 3 dB down; at 2ω_(b) it is −18dB (0.12), and at 4ω_(b) it is −36 dB (0.015) down. Higher-orderButterworth filters have a steeper roll-off, but most of their poles arecomplex and may affect negatively the control-loop's root locus.

G. Zero Offset Compensation

As noted above, zero offset may be introduced into a sensor voltagesignal by drift in the pre-amplification circuitry and by theanalog-to-digital converter. Slight differences in the pre-amplificationgains for positive and negative voltages due to the use of differentialcircuitry may worsen the zero offset effect. The errors vary betweentransmitters, and with transmitter temperature and component wear.

Audio quality (i.e., relatively low cost) analog-to-digital convertersmay be employed for economic reasons. These devices are not designedwith DC offset and amplitude stability as high priorities. FIGS. 19A-19Dshow how offset and positive and negative gains vary with chip operatingtemperature for one such converter (the AD1879 converter). Therepeatability of the illustrated trends is poor, and even allowing fortemperature compensation based on the trends, residual zero offset andpositive/negative gain mismatch remain.

If phase is calculated using the time difference between zero crossingpoints on the two sensor voltages, DC offset may lead to phase errors.This effect is illustrated by FIGS. 20A-20C. Each graph shows thecalculated phase offset as measured by the digital transmitter when thetrue phase offset is zero (i.e., at zero flow).

FIG. 20A shows phase calculated based on whole cycles starting withpositive zero-crossings. The mean value is 0.00627 degrees.

FIG. 20B shows phase calculated starting with negative zero-crossings.The mean value is 0.0109 degrees.

FIG. 20C shows phase calculated every half-cycle. FIG. 20C interleavesthe data from FIGS. 20A and 20B. The average phase (−0.00234) is closerto zero than in FIGS. 20A and 20B, but the standard deviation of thesignal is about six times higher.

More sophisticated phase measurement techniques, such as those based onFourier methods, are immune to DC offset. However, it is desirable toeliminate zero offset even when those techniques are used, since data isprocessed in whole-cycle packets delineated by zero crossing points.This allows simpler analysis of the effects of, for example, amplitudemodulation on apparent phase and frequency. In addition, gain mismatchbetween positive and negative voltages will introduce errors into anymeasurement technique.

The zero-crossing technique of phase detection may be used todemonstrate the impact of zero offset and gain mismatch error, and theirconsequent removal. FIGS. 21A and 21B illustrate the long term drift inphase with zero flow. Each point represents an average over one minuteof live data. FIG. 21A shows the average phase, and FIG. 21B shows thestandard deviation in phase. Over several hours, the drift issignificant. Thus, even if the meter were zeroed every day, which inmany applications would be considered an excessive maintenancerequirement, there would still be considerable phase drift.

1. Compensation Technique

A technique for dealing with voltage offset and gain mismatch uses thecomputational capabilities of the digital transmitter and does notrequire a zero flow condition. The technique uses a set of calculationseach cycle which, when averaged over a reasonable period (e.g., 10,000cycles), and excluding regions of major change (e.g., set point change,onset of aeration), converge on the desired zero offset and gainmismatch compensations.

Assuming the presence of up to three higher harmonics, the desiredwaveform for a sensor voltage SV(t) is of the form:SV(t)=A ₁ sin(ωt)+A ₂ sin(2ωt)+A ₃ sin(3ωt)+A ₄ sin(4ωt)

where A₁ designates the amplitude of the fundamental frequency componentand A₂-A₄ designate the amplitudes of the three harmonic components.However, in practice, the actual waveform is adulterated with zerooffset Z_(o) (which has a value close to zero) and mismatch between thenegative and positive gains G_(n) and G_(p). Without any loss ofgenerality, it can be assumed that G_(p) equals one and that G_(n) isgiven by:G _(n)=1+Ë _(G),  i.

where Ë_(G) represents the gain mismatch.

The technique assumes that the amplitudes A₁ and the frequency w areconstant. This is justified because estimates of Z_(o) and Ë_(G) arebased on averages taken over many cycles (e.g., 10,000 interleavedcycles occurring in about 1 minute of operation). When implementing thetechnique, the controller tests for the presence of significant changesin frequency and amplitude to ensure the validity of the analysis. Thepresence of the higher harmonics leads to the use of Fourier techniquesfor extracting phase and amplitude information for specific harmonics.This entails integrating SV(t) and multiplying by a modulating sine orcosine function.

The zero offset impacts the integral limits, as well as the functionalform. Because there is a zero offset, the starting point for calculationof amplitude and phase will not be at the zero phase point of theperiodic waveform SV(t). For zero offset Z_(o), the corresponding phaseoffset is, approximately,

$\varphi_{Z_{0}} = {- {\sin( \frac{Z_{0}}{A_{1}} )}}$

For small phase,

${\varphi_{Z_{o}} = {- \frac{Z_{o}}{A_{1}}}},$with corresponding time delay

$t_{Zo} = {\frac{\varphi_{Z_{o}}}{\omega}.}$

The integrals are scaled so that the limiting value (i.e., as Z_(o) andω_(G) approach zero) equals the amplitude of the relevant harmonic. Thefirst two integrals of interest are:

${I_{1{Ps}} = {\frac{2\omega}{\pi}{\int_{t_{z_{0}}}^{\frac{\pi}{\omega} + t_{z_{0}}}{{( {{S\;{V(t)}} + Z_{0}} ) \cdot {\sin\lbrack {\omega( {t - t_{Z_{0}}} )} \rbrack}}{\mathbb{d}t}}}}},{and}$$I_{1{Ns}} = {\frac{2\omega}{\pi}( {1 + ɛ_{G}} ){\int_{\frac{\pi}{\omega}t_{z_{0}}}^{{2\frac{\pi}{\omega}} + t_{z_{0}}}{{( {{{SV}(t)} + Z_{0}} ) \cdot {\sin\lbrack {\omega( {t - t_{Z_{0}}} )} \rbrack}}{{\mathbb{d}t}.}}}}$

These integrals represent what in practice is calculated during a normalFourier analysis of the sensor voltage data. The subscript 1 indicatesthe first harmonic, N and P indicate, respectively, the negative orpositive half cycle, and s and c indicate, respectively, whether a sineor a cosine modulating function has been used.

Strictly speaking, the mid-zero crossing point, and hence thecorresponding integral limits, should be given by π/ω−t_(Zo), ratherthan π/ω+t_(Zo). However, the use of the exact mid-point rather than theexact zero crossing point leads to an easier analysis, and betternumerical behavior (due principally to errors in the location of thezero crossing point). The only error introduced by using the exactmid-point is that a small section of each of the above integrals ismultiplied by the wrong gain (1 instead of 1+ε_(G) and vice versa).However, these errors are of order Z_(o) ²ε_(G) and are considerednegligible.

Using computer algebra and assuming small Z_(o) and ε_(G), first orderestimates for the integrals may be derived as:

${I_{1{Ps\_ est}} = {A_{1} + {\frac{4}{\pi}{Z_{0}\lbrack {1 + {\frac{2}{3}\frac{A_{2}}{A_{1}}} + {\frac{4}{15}\frac{A_{4}}{A_{1}}}} \rbrack}}}},{and}$$I_{1{Ns\_ est}} = {{( {1 + ɛ_{G}} )\lbrack {A_{1} - {\frac{4}{\pi}{Z_{0}\lbrack {1 + {\frac{2}{3}\frac{A_{2}}{A_{1}}} + {\frac{4}{15}\frac{A_{4}}{A_{1}}}} \rbrack}}} \rbrack}.}$

Useful related functions including the sum, difference, and ratio of theintegrals and their estimates may be determined. The sum of theintegrals may be expressed as:Sum _(1s)=(I _(1Ps) +I _(1Ns)),

while the sum of the estimates equals:

${Sum}_{1{s\_ est}} = {{A_{1}( {2 + ɛ_{G}} )} - {\frac{4}{\pi}Z_{0}{{ɛ_{G}\lbrack {1 + {\frac{2}{3}\frac{A_{2}}{A_{1}}} + {\frac{4}{15}\frac{A_{4}}{A_{1}}}} \rbrack}.}}}$

Similarly, the difference of the integrals may be expressed as:Diff_(1s) =I _(1Ps) −I _(1Ns),

while the difference of the estimates is:

${Diff}_{1{s\_ est}} = {{A_{1}ɛ_{G}} + {\frac{4}{\pi}{{{Z_{0}( {2 + ɛ_{G}} )}\lbrack {1 + {\frac{2}{3}\frac{A_{2}}{A_{1}}} + {\frac{4}{15}\frac{A_{4}}{A_{1}}}} \rbrack}.}}}$

Finally, the ratio of the integrals is:

${{Ratio}_{1s} = \frac{I_{1{Ps}}}{I_{1{Ns}}}},$

while the ratio of the estimates is:

${Ratio}_{1{s\_ est}} = {{\frac{1}{1 + ɛ_{G}}\lbrack {1 + {Z_{0}\lbrack {\frac{8}{15}\frac{{15A_{1}} + {10A_{2}} + {4A_{4}}}{\pi\; A_{1}^{2}}} \rbrack}} \rbrack}.}$

Corresponding cosine integrals are defined as:

${I_{1{Pc}} = {\frac{2\omega}{\pi}{\int_{t_{Z_{0}}}^{\frac{\pi}{\omega} + t_{Z_{0}}}{( {{S\;{V(t)}} + Z_{0}} ){\cos\lbrack {\omega( {t - t_{z_{0}}} )} \rbrack}{\mathbb{d}t}}}}},{and}$${I_{1{Nc}} = {\frac{2\omega}{\pi}( {1 + ɛ_{G}} ){\int_{\frac{\pi}{\omega} + t_{Z_{0}}}^{\frac{2\pi}{\omega} + t_{Z_{0}}}{( {{S\;{V(t)}} + Z_{0}} ){\cos\lbrack {\omega( {t - t_{z_{0}}} )} \rbrack}{\mathbb{d}t}}}}},$with estimates:

${I_{Pc\_ est} = {{- Z_{0}} + \frac{{40A_{2}} + {16A_{4}}}{15\pi}}},{and}$${I_{1{Nc\_ est}} = {( {1 + ɛ_{G}} )\lbrack {Z_{0} + \frac{{40A_{2}} + {16A_{4}}}{15\pi}} \rbrack}},$and sums:

${{Sum}_{1C} = {I_{1{Pc}} + I_{1{Nc}}}},{{{and}\mspace{14mu}{Sum}_{1{C\_ est}}} = {{ɛ_{G}\lbrack {Z_{0} + \frac{{40A_{2}} + {16A_{4}}}{15\pi}} \rbrack}.}}$Second harmonic integrals are:

${I_{2{Ps}} = {\frac{2\omega}{\pi}{\int_{t_{z_{0}}}^{\frac{\pi}{\omega}t_{z_{0}}}{( {{S\;{V(t)}} + Z_{0}} ){\sin\lbrack {2{\omega( {t - t_{z_{0}}} )}} \rbrack}{\mathbb{d}t}}}}},{and}$$I_{2{Ns}} = {\frac{2\omega}{\pi}( {1 + ɛ_{G}} ){\int_{\frac{\pi}{\omega} + t_{z_{0}}}^{\frac{2\pi}{\omega} + t_{z_{0}}}( {{{S\;{V(t)}} + {Z_{o}{\sin\lbrack {2{\omega( {t - t_{z_{0}}} )}} \rbrack}{\mathbb{d}t}}},} }}$with estimates:

${I_{2{Ps\_ est}} = {A_{2} + {\frac{8}{15\pi}{Z_{0}\lbrack {{- 5} + {9\frac{A_{3}}{A_{1}}}} \rbrack}}}},{and}$${I_{2{Ps\_ est}} = {( {1 + ɛ_{G}} )\lbrack {A_{2} - {\frac{8}{15\pi}{Z_{0}\lbrack {{- 5} + {9\frac{A_{3}}{A_{1}}}} \rbrack}}} \rbrack}},$and sums:

Sum_(2_(s)) = I_(2Ps) + I_(2Ns), and${Sum}_{2{{Ps}{\_ est}}} = {{A_{2}( {2 + ɛ_{G}} )} - {\frac{8}{15\pi}ɛ_{G}{{Z_{0}\lbrack {{- 5} + {9\frac{A_{3}}{A_{1}}}} \rbrack}.}}}$

The integrals can be calculated numerically every cycle. As discussedbelow, the equations estimating the values of the integrals in terms ofvarious amplitudes and the zero offset and gain values are rearranged togive estimates of the zero offset and gain terms based on the calculatedintegrals.

2. Example

The accuracy of the estimation equations may be illustrated with anexample. For each basic integral, three values are provided: the “true”value of the integral (calculated within Mathcad using Rombergintegration), the value using the estimation equation, and the valuecalculated by the digital transmitter operating in simulation mode,using Simpson's method with end correction.

Thus, for example, the value for I_(1Ps) calculated according to:

$I_{1{Ps}} = {\frac{2\omega}{\pi}{\int_{t_{z_{0}}}^{\frac{\pi}{\omega} + t_{z_{0}}}{( {{S\;{V(t)}} + Z_{0}} ){\sin\lbrack {\omega( {t - t_{z_{0}}} )} \rbrack}{\mathbb{d}t}}}}$

is 0.101353, while the estimated value (I_(1Ps) _(—) _(est)) calculatedas:

$I_{1{Ps\_ est}} = {A_{1} + {\frac{4}{\pi}{Z_{0}\lbrack {1 + {\frac{2}{3}\frac{A_{2}}{A_{1}}} + {\frac{4}{15}\frac{A_{4}}{A_{1}}}} \rbrack}}}$

is 0.101358. The value calculated using the digital transmitter insimulation mode is 0.101340. These calculations use the parameter valuesillustrated in Table C.

TABLE C Parameter Value Comment Ò 60π This corresponds to frequency = 80Hz, a typical value. A₁ .1 This more typically is 0.3, but it could besmaller with aeration. A₂ .01 This more typically is 0.005, but it couldbe larger with aeration. A₃ and A₄ .0 The digital Coriolis simulationmode only offers two harmonics, so these higher harmonics are ignored.However, they are small (<0.002). Z_(o) .001 Experience suggests thatthis is a large value for zero offset. Ë_(G) .001 Experience suggeststhis is a large value for gain mismatch.

The exact, estimate and simulation results from using these parametervalues are illustrated in Table D.

TABLE D Digital Coriolis Integral ‘Exact’ Value Estimate SimulationI_(1Ps) 0.101353 0.101358 0.101340 I_(1Ns) 0.098735 0.098740 0.098751I_(1Pc) 0.007487 0.007488 0.007500 I_(1Nc) −0.009496 −0.009498 −0.009531I_(2Ps) 0.009149 0.009151 0.009118 I_(2Ns) 0.010857 0.010859 0.010885

Thus, at least for the particular values selected, the estimates givenby the first order equations are extremely accurate. As Z_(o) and ε_(G)approach zero, the errors in both the estimate and the simulationapproach zero.

3. Implementation

The first order estimates for the integrals define a series ofnon-linear equations in terms of the amplitudes of the harmonics, thezero offset, and the gain mismatch. As the equations are non-linear, anexact solution is not readily available. However, an approximationfollowed by corrective iterations provides reasonable convergence withlimited computational overhead.

Conduit-specific ratios may be assumed for A₁-A₄. As such, no attempt ismade to calculate all of the amplitudes A₁-A₄. Instead, only A₁ and A₂are estimated using the integral equations defined above. Based onexperience of the relative amplitudes, A₃ may be approximated as A₂/2,and A₄ may be approximated as A₂/10.

The zero offset compensation technique may be implemented according tothe procedure 2200 illustrated in FIG. 22. During each cycle, thecontroller calculates the integrals I_(1ps), I_(1Ns), I_(1Pc), I_(1Nc),I_(2Ps), I_(2Ns) and related functions sum_(1s), ratio_(1s), sum_(1c)and sum_(2s) (step 2205). This requires minimal additional calculationbeyond the conventional Fourier calculations used to determinefrequency, amplitude and phase.

Every 10,000 cycles, the controller checks on the slope of the sensorvoltage amplitude A₁, using a conventional rate-of-change estimationtechnique (step 2210). If the amplitude is constant (step 2215), thenthe controller proceeds with calculations for zero offset and gainmismatch. This check may be extended to test for frequency stability.

To perform the calculations, the controller generates average values forthe functions (e.g., sum_(1s)) over the last 10,000 cycles. Thecontroller then makes a first estimation of zero offset and gainmismatch (step 2225):Z ₀ =−Sum _(1c)/2, andË _(G)=1/Ratio_(1s)−1

Using these values, the controller calculates an inverse gain factor (k)and amplitude factor (amp_factor) (step 2230):k=1.0/(1.0+0.5*Ë _(G)), andamp_factor=1+50/75*Sum _(2s)/Sum_(1s)

The controller uses the inverse gain factor and amplitude factor to makea first estimation of the amplitudes (step 2235):A ₁ =k*[Sum_(1s)/2+2/π*Z _(o)*ε_(G)*amp_factor], and  1.A ₂ =k*[Sum_(2s)/2−4/(3*π)*Z _(o)*ε_(G)  a.

The controller then improves the estimate by the following calculations,iterating as required (step 2240):x ₁ Z _(o),x ₂=ε_(G),ε_(G)=[1+8/π*x ₁ /A ₁*amp_factor]/Ratio_(1s)−1.0,Z _(o)=−Sum_(1c)/2+x ₂*(x ₁+2.773/π*A ₂)/2,A ₁ =k*[Sum_(1s)/2+2/π*x ₁ *x ₂*amp_factor],A ₂ =k*[Sum_(2s)/2−4/(15*π)*x ₁ *x ₂*(5−4.5*A ₂)].

The controller uses standard techniques to test for convergence of thevalues of Z_(o) and ε_(G). In practice the corrections are small afterthe first iteration, and experience suggests that three iterations areadequate.

Finally, the controller adjusts the raw data to eliminate Z_(o) andε_(G) (step 2245). The controller then repeats the procedure. Once zerooffset and gain mismatch have been eliminated from the raw data, thefunctions (i.e., sum_(1s)) used in generating subsequent values forZ_(o) and ε_(G) are based on corrected data. Accordingly, thesesubsequent values for Z_(o) and ε_(G) reflect residual zero offset andgain mismatch, and are summed with previously generated values toproduce the actual zero offset and gain mismatch. In one approach toadjusting the raw data, the controller generates adjustment parameters(e.g., S1_off and S2_off) that are used in converting the analog signalsfrom the sensors to digital data.

FIGS. 23A-23C, 24A and 24B show results obtained using the procedure2200. The short-term behavior is illustrated in FIGS. 23A-23C. Thisshows consecutive phase estimates obtained five minutes after startup toallow time for the procedure to begin affecting the output. Phase isshown based on positive zero-crossings, negative zero-crossings, andboth.

The difference between the positive and negative mean values has beenreduced by a factor of 20, with a corresponding reduction in mean zerooffset in the interleaved data set. The corresponding standard deviationhas been reduced by a factor of approximately 6.

Longer term behavior is shown in FIGS. 24A and 24B. The initial largezero offset is rapidly corrected, and then the phase offset is keptclose to zero over many hours. The average phase offset, excluding thefirst few values, is 6.14e⁻⁶, which strongly suggests that the procedureis successful in compensating for changes in voltage offset and gainimbalance.

Typical values for Z_(o) and ε_(G) for the digital Coriolis meter areZ_(o)=−7.923e⁻⁴ and ε_(G)=−1.754e⁻⁵ for signal SV₁, and Z_(o)=−8.038e⁻⁴and ε_(G)=+6.93e⁻⁴ for signal SV₂.

H. Dynamic Analysis

In general, conventional measurement calculations for Coriolis metersassume that the frequency and amplitude of oscillation on each side ofthe conduit are constant, and that the frequency on each side of theconduit is identical and equal to the so-called resonant frequency.Phases generally are not measured separately for each side of theconduit, and the phase difference between the two sides is assumed to beconstant for the duration of the measurement process. Precisemeasurements of frequency, phase and amplitude every half-cycle usingthe digital meter demonstrate that these assumptions are only valid whenparameter values are averaged over a time period on the order ofseconds. Viewed at 100 Hz or higher frequencies, these parametersexhibit considerable variation. For example, during normal operation,the frequency and amplitude values of SV₁ may exhibit strong negativecorrelation with the corresponding SV₂ values. Accordingly, conventionalmeasurement algorithms are subject to noise attributable to thesedynamic variations. The noise becomes more significant as themeasurement calculation rate increases. Other noise terms may beintroduced by physical factors, such as flowtube dynamics, dynamicnon-linearities (e.g. flowtube stiffness varying with amplitude), or thedynamic consequences of the sensor voltages providing velocity datarather than absolute position data.

The described techniques exploit the high precision of the digital meterto monitor and compensate for dynamic conduit behavior to reduce noiseso as to provide more precise measurements of process variables such asmass flow and density. This is achieved by monitoring and compensatingfor such effects as the rates of change of frequency, phase andamplitude, flowtube dynamics, and dynamic physical non-idealities. Aphase difference calculation which does not assume the same frequency oneach side has already been described above. Other compensationtechniques are described below.

Monitoring and compensation for dynamic effects may take place at theindividual sensor level to provide corrected estimates of phase,frequency, amplitude or other parameters. Further compensation may alsotake place at the conduit level, where data from both sensors arecombined, for example in the calculation of phase difference and averagefrequency. These two levels may be used together to providecomprehensive compensation.

Thus, instantaneous mass flow and density measurements by the flowmetermay be improved by modeling and accounting for dynamic effects offlowmeter operation. In general, 80% or more of phase noise in aCoriolis flowmeter may be attributed to flowtube dynamics (sometimesreferred to as “ringing”), rather than to process conditions beingmeasured. The application of a dynamic model can reduce phase noise by afactor of 4 to 10, leading to significantly improved flow measurementperformance. A single model is effective for all flow rates andamplitudes of oscillation. Generally, computational requirements arenegligible.

The dynamic analysis may be performed on each of the sensor signals inisolation from the other. This avoids, or at least delays, modeling thedynamic interaction between the two sides of the conduit, which islikely to be far more complex than the dynamics at each sensor. Also,analyzing the individual sensor signals is more likely to be successfulin circumstances such as batch startup and aeration where the two sidesof the conduit are subject to different forces from the process fluid.

In general, the dynamic analysis considers the impact of time-varyingamplitude, frequency and phase on the calculated values for theseparameters. While the frequency and amplitude are easily defined for theindividual sensor voltages, phase is conventionally defined in terms ofthe difference between the sensor voltages. However, when a Fourieranalysis is used, phase for the individual sensor may be defined interms of the difference between the midpoint of the cycle and theaverage 180° phase point.

Three types of dynamic effects are measurement error and the so-called“feedback” and “velocity” effects. Measurement error results because thealgorithms for calculating amplitude and phase assume that frequency,amplitude, and phase are constant over the time interval of interest.Performance of the measurement algorithms may be improved by correctingfor variations in these parameters.

The feedback effect results from supplying energy to the conduit to makeup for energy loss from the conduit so as to maintain a constantamplitude of oscillation. The need to add energy to the conduit is onlyrecognized after the amplitude of oscillation begins to deviate from adesired setpoint. As a result, the damping term in the equation ofmotion for the oscillating conduit is not zero, and, instead, constantlydithers around zero. Although the natural frequency of the conduit doesnot change, it is obscured by shifts in the zero-crossings (i.e., phasevariations) associated with these small changes in amplitude.

The velocity effect results because the sensor voltages observe conduitvelocity, but are analyzed as being representative of conduit position.A consequence of this is that the rate of change of amplitude has animpact on the apparent frequency and phase, even if the true values ofthese parameters are constant.

1. Sensor-Level Compensation for Amplitude Modulation

One approach to correcting for dynamic effects monitors the amplitudesof the sensor signals and makes adjustments based on variations in theamplitudes. For purposes of analyzing dynamic effects, it is assumedthat estimates of phase, frequency and amplitude may be determined foreach sensor voltage during each cycle. As shown in FIG. 25, calculationsare based on complete but overlapping cycles. Each cycle starts at azero crossing point, halfway through the previous cycle. Positive cyclesbegin with positive voltages immediately after the initialzero-crossing, while negative cycles begin with negative voltages. Thuscycle n is positive, while cycles n−1 and n+1 are negative. It isassumed that zero offset correction has been performed so that zerooffset is negligible. It also is assumed that higher harmonics may bepresent.

Linear variation in amplitude, frequency, and phase are assumed. Underthis assumption, the average value of each parameter during a cycleequals the instantaneous value of the parameter at the mid-point of thecycle. Since the cycles overlap by 180 degrees, the average value for acycle equals the starting value for the next cycle.

For example, cycle n is from time 0 to 2π/ω. The average values ofamplitude, frequency and phase equal the instantaneous values at themid-point, π/ω, which is also the starting point for cycle n+1, which isfrom time π/ω to 3π/ω. Of course, these timings are approximate, since ωalso varies with time.

a. Dynamic Effect Compensation Procedure

The controller accounts for dynamic effects according to the procedure2600 illustrated in FIG. 26. First, the controller produces a frequencyestimate (step 2605) by using the zero crossings to measure the timebetween the start and end of the cycle, as described above. Assumingthat frequency varies linearly, this estimate equals the time-averagedfrequency over the period.

The controller then uses the estimated frequency to generate a firstestimate of amplitude and phase using the Fourier method described above(step 2610). As noted above, this method eliminates the effects ofhigher harmonics.

Phase is interpreted in the context of a single waveform as being thedifference between the start of the cycle (i.e., the zero-crossingpoint) and the point of zero phase for the component of SV(t) offrequency ω, expressed as a phase offset. Since the phase offset is anaverage over the entire waveform, it may be used as the phase offsetfrom the midpoint of the cycle. Ideally, with no zero offset andconstant amplitude of oscillation, the phase offset should be zero everycycle. In practice, however, it shows a high level of variation andprovides an excellent basis for correcting mass flow to account fordynamic changes in amplitude.

The controller then calculates a phase difference (step 2615). Though anumber of definitions of phase difference are possible, the analysisassumes that the average phase and frequency of each sensor signal isrepresentative of the entire waveform. Since these frequencies aredifferent for SV₁ and SV₂, the corresponding phases are scaled to theaverage frequency. In addition, the phases are shifted to the samestarting point (i.e., the midpoint of the cycle on SV₁). After scaling,they are subtracted to provide the phase difference.

The controller next determines the rate of change of the amplitude forthe cycle n (step 2620):

$\begin{matrix}{{roc\_ amp}_{n} \approx \frac{{{amp}( {{end}\mspace{14mu}{of}\mspace{14mu}{cycle}} )} - {{amp}( {{start}\mspace{14mu}{of}\mspace{14mu}{cycle}} )}}{{period}\mspace{14mu}{of}\mspace{14mu}{cycle}}} \\{= {( {{amp}_{n + 1} - {amp}_{n - 1}} ){{freq}_{n}.}}}\end{matrix}$

This calculation assumes that the amplitude from cycle n+1 is availablewhen calculating the rate of change of cycle n. This is possible if thecorrections are made one cycle after the raw amplitude calculations havebeen made. The advantage of having an accurate estimate of the rate ofchange, and hence good measurement correction, outweighs the delay inthe provision of the corrected measurements, which, in oneimplementation, is on the order of 5 milliseconds. The most recentlygenerated information is always used for control of the conduit (i.e.,for generation of the drive signal).

If desired, a revised estimate of the rate of change can be calculatedafter amplitude correction has been applied (as described below). Thisresults in iteration to convergence for the best values of amplitude andrate of change.

b. Frequency Compensation for Feedback and Velocity Effects

As noted above, the dynamic aspects of the feedback loop introduce timevarying shifts in the phase due to the small deviations in amplitudeabout the set-point. This results in the measured frequency, which isbased on zero-crossings, differing from the natural frequency of theconduit. If velocity sensors are used, an additional shift in phaseoccurs. This additional shift is also associated with changes in thepositional amplitude of the conduit. A dynamic analysis can monitor andcompensate for these effects. Accordingly, the controller uses thecalculated rate of amplitude change to correct the frequency estimate(step 2625).

The position of an oscillating conduit in a feedback loop that isemployed to maintain the amplitude of oscillation of the conduitconstant may be expressed as:X=A(t)sin(ω₀ t−θ(t)),

where θ(t) is the phase delay caused by the feedback effect. Themechanical Q of the oscillating conduit is typically on the order of1000, which implies small deviations in amplitude and phase. Under theseconditions, θ(t) is given by:

${\theta(t)} \approx {- {\frac{\overset{.}{A}(t)}{2\omega_{0}{A(t)}}.}}$Since each sensor measures velocity:

$\begin{matrix}{{S\;{V(t)}} = {\overset{.}{X}(t)}} \\{= {{{\overset{.}{A}(t)}{\sin\lbrack {{\omega_{0}t} - {\theta(t)}} \rbrack}} + {\lbrack {\omega_{0} - {\overset{.}{\theta}(t)}} \rbrack{A(t)}{\cos\lbrack {{\omega_{0}t} - {\theta(t)}} \rbrack}}}} \\{{= {\omega_{0}{{A(t)}\lbrack {( {1 - \frac{\theta}{\omega_{0}}} )^{2} + ( \frac{\overset{.}{A}(t)}{\omega_{0}{A(t)}} )^{2}} \rbrack}^{1/2}{\cos( {{\omega_{0}t} - {\theta(t)} - {\gamma(t)}} )}}},}\end{matrix}$where Υ(t) is the phase delay caused by the velocity effect:

${\gamma(t)} = {{\tan^{- 1}( \frac{\overset{.}{A}(t)}{\omega_{0}{A(t)}( {1 - \frac{\overset{.}{\theta}}{\omega_{0}}} )} )}.}$

Since the mechanical Q of the conduit is typically on the order of 1000,and, hence, variations in amplitude and phase are small, it isreasonable to assume:

$\frac{\overset{.}{\theta}}{\omega_{0}}{\operatorname{<<}1}\mspace{14mu}{and}\mspace{14mu}\frac{\overset{.}{A}(t)}{\omega_{0}{A(t)}}{\operatorname{<<}1.}$

This means that the expression for SV(t) may be simplified to:SV(t)≈ω₀ A(t)cos(ω_(h) t−θ(t)−γ(t)),

and for the same reasons, the expression for the velocity offset phasedelay may be simplified to:

${\gamma(t)} \approx {\frac{\overset{.}{A}(t)}{\omega_{0}{A(t)}}.}$

Summing the feedback and velocity effect phase delays gives the totalphase delay:

${{\varphi(t)} = {{{{\theta(t)} + {\gamma(t)}} \approx {{- \frac{\overset{.}{A}(t)}{2\omega_{0}{A(t)}}} + \frac{\overset{.}{A}(t)}{w_{0}{A(t)}}}} = \frac{\overset{.}{A}(t)}{2\omega_{0}{A(t)}}}},$

and the following expression for SV(t):SV(t)≈ω₀ A(t)cos [ω₀ t−φ(t)].

From this, the actual frequency of oscillation may be distinguished fromthe natural frequency of oscillation. Though the former is observed, thelatter is useful for density calculations. Over any reasonable length oftime, and assuming adequate amplitude control, the averages of these twofrequencies are the same (because the average rate of change ofamplitude must be zero). However, for improved instantaneous densitymeasurement, it is desirable to compensate the actual frequency ofoscillation for dynamic effects to obtain the natural frequency. This isparticularly useful in dealing with aerated fluids for which theinstantaneous density can vary rapidly with time.

The apparent frequency observed for cycle n is delineated by zerocrossings occurring at the midpoints of cycles n−1 and n+1. The phasedelay due to velocity change will have an impact on the apparent startand end of the cycle:

$\begin{matrix}{{obs\_ freq}_{n} = {{obs\_ freq}_{n - 1} + {\frac{{true\_ freq}_{n}}{2\pi}( {\varphi_{n + 1} - \varphi_{n - 1}} )}}} \\{= {{obs\_ freq}_{n - 1} + {\frac{{true\_ freq}_{n - 1}}{2\pi}\begin{pmatrix}{\frac{{\overset{.}{A}}_{n + 1}}{4\pi\mspace{14mu}{true\_ freq}_{n}A_{n + 1}} -} \\\frac{{\overset{.}{A}}_{n - 1}}{4\pi\mspace{14mu}{true\_ freq}_{n}A_{n - 1}}\end{pmatrix}}}} \\{= {{obs\_ freq}_{n - 1} + {\frac{1}{8\pi^{2}}{( {\frac{{\overset{.}{A}}_{n + 1}}{A_{n + 1}} - \frac{{\overset{.}{A}}_{n - 1}}{A_{n - 1}}} ).}}}}\end{matrix}$

Based on this analysis, a correction can be applied using an integratederror term:

${{error\_ sum}_{n} = {{error\_ sum}_{n - 1} - {\frac{1}{8\pi^{2}}( {\frac{{\overset{.}{A}}_{n + 1}}{A_{n + 1}} - \frac{{\overset{.}{A}}_{n - 1}}{A_{n - 1}}} )}}},$andest_freq_(n)=obs_freq_(n)−error_sum_(n),where the value of error_sum at startup (i.e., the value at cycle zero)is:

${error\_ sum}_{0} = {{- \frac{1}{8\pi^{2}}}{( {\frac{{\overset{.}{A}}_{0}}{A_{0}} + \frac{{\overset{.}{A}}_{1}}{A_{1}}} ).}}$

Though these equations include a constant term having a value of ⅛π²,actual data has indicated that a constant term of ⅛π is moreappropriate. This discrepancy may be due to unmodeled dynamics that maybe resolved through further analysis.

The calculations discussed above assume that the true amplitude ofoscillation, A, is available. However, in practice, only the sensorvoltage SV is observed. This sensor voltage may be expressed as:SV(t)≈ω₀ {dot over (A)}(t)cos(ω₀ t−φ(t))

The amplitude, amp_SV(t), of this expression is:amp_(—) SV(t)≈ω₀ {dot over (A)}(t).

The rate of change of this amplitude is:roc_amp_(—) SV(t)≈ω₀ {dot over (A)}(t)

so that the following estimation can be used:

$\frac{\overset{.}{A}(t)}{A(t)} \approx {\frac{{roc\_ amp}{\_ SV}(t)}{{amp\_ SV}(t)}.}$

c. Application of Feedback and Velocity Effect Frequency Compensation

FIGS. 27A-32B illustrate how application of the procedure 2600 improvesthe estimate of the natural frequency, and hence the process density,for real data from a meter having a one inch diameter conduit. Each ofthe figures shows 10,000 samples, which are collected in just over 1minute.

FIGS. 27A and 27B show amplitude and frequency data from SV₁, taken whenrandom changes to the amplitude set-point have been applied. Since theconduit is full of water and there is no flow, the natural frequency isconstant. However, the observed frequency varies considerably inresponse to changes in amplitude. The mean frequency value is 81.41 Hz,with a standard deviation of 0.057 Hz.

FIGS. 28A and 28B show, respectively, the variation in frequency fromthe mean value, and the correction term generated using the procedure2600. The gross deviations are extremely well matched. However, there isadditional variance in frequency which is not attributable to amplitudevariation. Another important feature illustrated by FIG. 28B is that theaverage is close to zero as a result of the proper initialization of theerror term, as described above.

FIGS. 29A and 92B compare the raw frequency data (FIG. 29A) with theresults of applying the correction function (FIG. 29B). There has been anegligible shift in the mean frequency, while the standard deviation hasbeen reduced by a factor of 4.4. From FIG. 29B, it is apparent thatthere is residual structure in the corrected frequency data. It isexpected that further analysis, based on the change in phase across acycle and its impact on the observed frequency, will yield further noisereductions.

FIGS. 30A and 30B show the corresponding effect on the averagefrequency, which is the mean of the instantaneous sensor voltagefrequencies. Since the mean frequency is used to calculate the densityof the process fluid, the noise reduction (here by a factor of 5.2) willbe propagated into the calculation of density.

FIGS. 31A and 31B illustrate the raw and corrected average frequency fora 2″ diameter conduit subject to a random amplitude set-point. The 2″flowtube exhibits less frequency variation that the 1″, for both raw andcorrected data. The noise reduction factor is 4.0.

FIGS. 32A and 32B show more typical results with real flow data for theone inch flowtube. The random setpoint algorithm has been replaced bythe normal constant setpoint. As a result, there is less amplitudevariation than in the previous examples, which results in a smallernoise reduction factor of 1.5.

d. Compensation of Phase Measurement for Amplitude Modulation

Referring again to FIG. 26, the controller next compensates the phasemeasurement to account for amplitude modulation assuming the phasecalculation provided above (step 2630). The Fourier calculations ofphase described above assume that the amplitude of oscillation isconstant throughout the cycle of data on which the calculations takeplace. This section describes a correction which assumes a linearvariation in amplitude over the cycle of data.

Ignoring higher harmonics, and assuming that any zero offset has beeneliminated, the expression for the sensor voltage is given by:SV(t)≈A ₁(1+λ_(A) t)sin(ωt)where Î_(A) is a constant corresponding to the relative change inamplitude with time. As discussed above, the integrals I₁ and I₂ may beexpressed as:

${I_{1} = {\frac{2\;\omega}{\pi}{\int_{0}^{- \frac{2\;\pi}{\omega}}{{{SV}(t)}{\sin( {\omega\; t} )}\ {\mathbb{d}t}}}}},{{{and}\mspace{14mu} I_{2}} = {\frac{2\;\omega}{\pi}{\int_{0}^{- \frac{2\;\pi}{\omega}}{{{SV}(t)}{\cos( {\omega\; t} )}\ {\mathbb{d}t}}}}},$Evaluating these integrals results in:

${I_{1} = {A_{1}( {1 + {\frac{\pi}{\omega}\lambda_{A}}} )}},{{{and}\mspace{14mu} I_{2}} = {A_{1}\frac{1}{2\omega}{\lambda_{A}.}}}$Substituting these expressions into the calculation for amplitude andexpanding as a series in Î_(A) results in:

${Amp} = {{A_{1}( {1 + {\frac{\pi}{\omega}\lambda_{A}} + {\frac{1}{8\omega^{2}}\lambda_{A}^{2}} + \ldots}\mspace{14mu} )}.}$

Assuming Î_(A) is small, and ignoring all terms after the first orderterm, this may be simplified to:

${Amp} = {{A_{1}( {1 + {\frac{\pi}{\omega}\lambda_{A}}} )}.}$This equals the amplitude of SV(t) at the midpoint of the cycle (t=π/ω).Accordingly, the amplitude calculation provides the required resultwithout correction.

For the phase calculation, it is assumed that the true phase differenceand frequency are constant, and that there is no voltage offset, whichmeans that the phase value should be zero. However, as a result ofamplitude modulation, the correction to be applied to the raw phase datato compensate for amplitude modulation is:

${Phase} = {{\tan^{- 1}( \frac{\lambda_{A}}{2( {{\pi\;\lambda_{A}} + \omega} )} )}.}$

Assuming that the expression in brackets is small, the inverse tangentfunction can be ignored.

A more elaborate analysis considers the effects of higher harmonics.Assuming that the sensor voltage may be expressed as:SV(t)=(1+λ_(A) t)[A ₁ sin(ωt)+A ₂ sin(2ωt)+A ₃ sin(ωt)+A ₄ sin(4ωt)]

such that all harmonic amplitudes increase at the same relative rateover the cycle, then the resulting integrals may be expressed as:

${I_{1} = {A_{1}( {1 + {\frac{\pi}{\omega}\lambda_{A}}} )}},{{and}\mspace{14mu} I_{2}\frac{- 1}{60\omega}{\lambda_{A}( {{30A_{1}} + {80A_{2}} + {45A_{3}} + {32A_{4}}} )}}$for positive cycles, and

$I_{2}\frac{- 1}{60\omega}{\lambda_{A}( {{30A_{1}} - {80A_{2}} + {45A_{3}} - {32A_{4}}} )}$

for negative cycles.

For amplitude, substituting these expressions into the calculationsestablishes that the amplitude calculation is only affected in thesecond order and higher terms, so that no correction is necessary to afirst order approximation of the amplitude. For phase, the correctionterm becomes:

$\frac{- 1}{60}{\lambda_{A}( \frac{{30A_{1}} + {80A_{2}} + {45A_{3}} + {32A_{4}}}{A_{1}( {{\pi\;\lambda_{A}} + \omega} )} )}$

for positive cycles, and

$\frac{- 1}{60}{\lambda_{A}( \frac{{30A_{1}} - {80A_{2}} + {45A_{3}} - {32A_{4}}}{A_{1}( {{\pi\lambda}_{A} + \omega} )} )}$for negative cycles. These correction terms assume the availability ofthe amplitudes of the higher harmonics. While these can be calculatedusing the usual Fourier technique, it is also possible to approximatesome or all them using assumed ratios between the harmonics. Forexample, for one implementation of a one inch diameter conduit, typicalamplitude ratios are A₁=1.0, A₂=0.01, A₃=0.005, and A₄=0.001.

e. Application of Amplitude Modulation Compensation to Phase

Simulations have been carried out using the digital transmitter,including the simulation of higher harmonics and amplitude modulation.One example uses f=80 Hz, A₁(t=0)=0.3, A₂=0, A₃=0, A₄=0, Î_(A)=1e⁻⁵*48KHz (sampling rate)=0.47622, which corresponds to a high rate of changeof amplitude, but with no higher harmonics. Theory suggests a phaseoffset of −0.02706 degrees. In simulation over 1000 cycles, the averageoffset is −0.02714 degrees, with a standard deviation of only 2.17e⁻⁶.The difference between simulation and theory (approx 0.3% of thesimulation error) is attributable to the model's assumption of a linearvariation in amplitude over each cycle, while the simulation generatesan exponential change in amplitude.

A second example includes a second harmonic, and has the parameters f=80Hz, A₁(t=0)=0.3, A₂(t=0)=0.003, A₃=0, A₄=0, Î_(A)=−1e⁻⁶*48 KHz (samplingrate)=−0.047622. For this example, theory predicts the phase offset tobe +2.706e⁻³, +/−2.66% for positive or negative cycles. In simulation,the results are 2.714e⁻³+/−2.66%, which again matches well.

FIGS. 33A-34B give examples of how this correction improves realflowmeter data. FIG. 33A shows raw phase data from SV₁, collected from a1″ diameter conduit, with low flow assumed to be reasonably constant.FIG. 33B shows the correction factor calculated using the formuladescribed above, while FIG. 33C shows the resulting corrected phase. Themost apparent feature is that the correction has increased the varianceof the phase signal, while still producing an overall reduction in thephase difference (i.e., SV₂−SV₁) standard deviation by a factor of 1.26,as shown in FIGS. 34A and 34B. The improved performance results becausethis correction improves the correlation between the two phases, leadingto reduced variation in the phase difference. The technique worksequally well in other flow conditions and on other conduit sizes.

f. Compensation to Phase Measurement for Velocity Effect

The phase measurement calculation is also affected by the velocityeffect. A highly effective and simple correction factor, in radians, isof the form

${{c_{v}( t_{k} )} = {\frac{1}{\pi}\Delta\;{{SV}( t_{k} )}}},$

where, SV(t_(k)) is the relative rate of change of amplitude and may beexpressed as:

${{\Delta\;{{SV}( t_{k} )}} = {\frac{{{SV}( t_{k + 1} )} - {{SV}( t_{k - 1} )}}{t_{k + 1} - t_{k - 1}} \cdot \frac{1}{{SV}( t_{k} )}}},$

where t_(k) is the completion time for the cycle for which ,SV(t_(k)) isbeing determined, t_(k+1) is the completion time for the next cycle, andt_(k−1) is the completion time of the previous cycle. ,SV is an estimateof the rate of change of SV, scaled by its absolute value, and is alsoreferred to as the proportional rate of change of SV.

FIGS. 35A-35E illustrate this technique. FIG. 35A shows the raw phasedata from a single sensor (SV₁), after having applied the amplitudemodulation corrections described above. FIG. 35B shows the correctionfactor in degrees calculated using the equation above, while FIG. 35Cshows the resulting corrected phase. It should be noted that thestandard deviation of the corrected phase has actually increasedrelative to the raw data. However, when the corresponding calculationstake place on the other sensor (SV₂), there is an increase in thenegative correlation (from −0.8 to −0.9) between the phases on the twosignals. As a consequence, the phase difference calculations based onthe raw phase measurements (FIG. 35D) have significantly more noise thanthe corrected phase measurements (FIG. 35E).

Comparison of FIGS. 35D and 35E shows the benefits of this noisereduction technique. It is immediately apparent from visual inspectionof FIG. 35E that the process variable is decreasing, and that there aresignificant cycles in the measurement, with the cycles beingattributable, perhaps, to a poorly conditioned pump. None of this isdiscernable from the uncorrected phase difference data of FIG. 35D.

g. Application of Sensor Level Noise Reduction

The combination of phase noise reduction techniques described aboveresults in substantial improvements in instantaneous phase differencemeasurement in a variety of flow conditions, as illustrated in FIGS.36A-36L. Each graph shows three phase difference measurements calculatedsimultaneously in real time by the digital Coriolis transmitteroperating on a one inch conduit. The middle band 3600 shows phase datacalculated using the simple time-difference technique. The outermostband 3605 shows phase data calculated using the Fourier-based techniquedescribed above.

It is perhaps surprising that the Fourier technique, which uses far moredata, a more sophisticated analysis, and much more computational effort,results in a noisier calculation. This can be attributed to thesensitivity of the Fourier technique to the dynamic effects describedabove. The innermost band of data 3610 shows the same Fourier data afterthe application of the sensor-level noise reduction techniques. As canbe seen, substantial noise reduction occurs in each case, as indicatedby the standard deviation values presented on each graph.

FIG. 36A illustrates measurements with no flow, a full conduit, and nopump noise. FIG. 36B illustrates measurements with no flow, a fullconduit, and the pumps on. FIG. 36C illustrates measurements with anempty, wet conduit. FIG. 36D illustrates measurements at a low flowrate. FIG. 36E illustrates measurements at a high flow rate. FIG. 36Fillustrates measurements at a high flow rate and an amplitude ofoscillation of 0.03V. FIG. 36G illustrates measurements at a low flowrate with low aeration. FIG. 36H illustrates measurements at a low flowrate with high aeration. FIG. 36I illustrates measurements at a highflow rate with low aeration. FIG. 36J illustrates measurements at a highflow rate with high aeration. FIG. 36K illustrates measurements for anempty to high flow rate transition. FIG. 36L illustrates measurementsfor a high flow rate to empty transition.

2. Flowtube Level Dynamic Modeling

A dynamic model may be incorporated in two basic stages. In the firststage, the model is created using the techniques of systemidentification. The flowtube is “stimulated” to manifest its dynamics,while the true mass flow and density values are kept constant. Theresponse of the flowtube is measured and used in generating the dynamicmodel. In the second stage, the model is applied to normal flow data.Predictions of the effects of flowtube dynamics are made for both phaseand frequency. The predictions then are subtracted from the observeddata to leave the residual phase and frequency, which should be due tothe process alone. Each stage is described in more detail below.

a. System Identification

System identification begins with a flowtube full of water, with noflow. The amplitude of oscillation, which normally is kept constant, isallowed to vary by assigning a random setpoint between 0.05 V and 0.3 V,where 0.3 V is the usual value. The resulting sensor voltages are shownin FIG. 37A, while FIGS. 37B and 37C show, respectively, thecorresponding calculated phase and frequency values. These values arecalculated once per cycle. Both phase and frequency show a high degreeof “structure.” Since the phase and frequency corresponding to mass floware constant, this structure is likely to be related to flowtubedynamics. Observable variables that will predict this structure when thetrue phase and frequency are not known to be constant may be expressedas set forth below.

First, as noted above, ,SV(t_(k)) may be expressed as:

${\Delta\;{{SV}( t_{k} )}} = {\frac{{{SV}( t_{k + 1} )} - {{SV}( t_{k - 1} )}}{t_{k + 1} - t_{k - 1}} \cdot {\frac{1}{{SV}( t_{k} )}.}}$

This expression may be used to determine ,SV₁ and ,SV₂.

The phase of the flowtube is related to ,, which is defined as ,SV₁−,SV₂, while the frequency is related to ,⁺, which is defined as ,SV₁+,SV₂. These parameters are illustrated in FIGS. 37D and 37E. ComparingFIG. 37B to FIG. 37D and FIG. 37C to FIG. 37E shows the strikingrelationship between ,⁻ and phase and between ,⁺ and frequency.

Some correction for flowtube dynamics may be obtained by subtracting amultiple of the appropriate prediction function from the phase and/orthe frequency. Improved results may be obtained using a model of theform:y(k)+a ₁ y(k−1)+ . . . +a _(n) y(k−n)=b ₀ u(k)+b ₁ u(k−1)+ . . . +b _(m)u(k−m),where y(k) is the output (i.e., phase or frequency) and u is theprediction function (i.e., ,⁻ or ,⁺). The technique of systemidentification suggests values for the orders n and m, and thecoefficients a_(i) and b_(j), of what are in effect polynomials in time.The value of y(k) can be calculated every cycle and subtracted from theobserved phase or frequency to get the residual process value.

It is important to appreciate that, even in the absence of dynamiccorrections, the digital flowmeter offers very good precision over along period of time. For example, when totalizing a batch of 200 kg, thedevice readily achieves a repeatability of less that 0.03%. The purposeof the dynamic modeling is to improve the dynamic precision. Thus, rawand compensated values should have similar mean values, but reductionsin “variance” or “standard deviation.”

FIGS. 38A and 39A show raw and corrected frequency values. The meanvalues are similar, but the standard deviation has been reduced by afactor of 3.25. Though the gross deviations in frequency have beeneliminated, significant “structure” remains in the residual noise. Thisstructure appears to be unrelated to the ,⁺ function. The model used isa simple first order model, where m=n=1.

FIGS. 38B and 39B show the corresponding phase correction. The meanvalue is minimally affected, while the standard deviation is reduced bya factor of 7.9. The model orders are n=2 and m=10. Some structureappears to remain in the residual noise. It is expected that thisstructure is due to insufficient excitation of the phase dynamics bysetpoint changes.

More effective phase identification has been achieved through furthersimulation of flowtube dynamics by continuous striking of the flowtubeduring data collection (set point changes are still carried out). FIGS.38C and 39C show the effects of correction under these conditions. Asshown, the standard deviation is reduced by a factor of 31. This moreeffective model is used in the following discussions.

b. Application to Flow Data

The real test of an identified model is the improvements it provides fornew data. At the outset, it is useful to note a number of observations.First, the mean phase, averaged over, for example, ten seconds or more,is already quite precise. In the examples shown, phase values areplotted at 82 Hz or thereabouts. The reported standard deviation wouldbe roughly ⅓ of the values shown when averaged to 10 Hz, and 1/9 whenaverages to 1 Hz. For reference, on a one inch flow tube, one degree ofphase difference corresponds to about 1 kg/s flow rate.

The expected benefit of the technique is that of providing a much betterdynamic response to true process changes, rather than improving averageprecision. Consequently, in the following examples, where the flow isnon-zero, small flow step changes are introduced every ten seconds orso, with the expectation that the corrected phase will show up the stepchanges more clearly.

FIGS. 38D and 39D show the correction applied to a full flowtube withzero flow, just after startup. The ring-down effect characteristic ofstartup is clearly evident in the raw data (FIG. 38D), but this iseliminated by the correction (FIG. 39D), leading to a standard deviationreduction of a factor of 23 over the whole data set. Note that thecorrected measurement closely resembles white noise, suggesting mostflowtube dynamics have been captured.

FIGS. 38E and 39E show the resulting correction for a “drained”flowtube. Noise is reduced by a factor of 6.5 or so. Note, however, thatthere appears to be some residual structure in the noise.

The effects of the technique on low (FIGS. 38F and 39F), medium (FIGS.38G and 39G), and high (FIGS. 38H and 39H) flow rates are alsoillustrated, each with step changes in flow every ten seconds. In eachcase, the pattern is the same: the corrected average flows (FIGS.39F-39H) are identical to the raw average flows (FIGS. 38F-38H), but thedynamic noise is reduced considerably. In FIG. 39H, this leads to theemergence of the step changes which previously had been submerged innoise (FIG. 38H).

3. Extensions of Dynamic Monitoring and Compensation Techniques

The previous sections have described a variety of techniques (physicalmodeling, system identification, heuristics) used to monitor andcompensate for different aspects of dynamic behavior (frequency andphase noise caused by amplitude modulation, velocity effect, flowtubedynamics at both the sensor and the flowtube level). By naturalextension, similar techniques well-known to practitioners of controland/or instrumentation, including those of artificial intelligence,neural networks, fuzzy logic, and genetic algorithms, as well asclassical modeling and identification methods, may be applied to theseand other aspects of the dynamic performance of the meter. Specifically,these might include monitoring and compensation for frequency, amplitudeand/or phase variation at the sensor level, as well as average frequencyand phase difference at the flowtube level, as these variations occurwithin each measurement interval, as well as the time betweenmeasurement intervals (where measurement intervals do not overlap).

This technique is unusual in providing both reduced noise and improveddynamic response to process measurement changes. As such, the techniquepromises to be highly valuable in the context of flow measurement.

I. Aeration

The digital flowmeter provides improved performance in the presence ofaeration in the conduit. Aeration causes energy losses in the conduitthat can have a substantial negative impact on the measurements producedby a mass flowmeter and can result in stalling of the conduit.Experiments have shown that the digital flowmeter has substantiallyimproved performance in the presence of aeration relative totraditional, analog flowmeters. This performance improvement may stemfrom the meter's ability to provide a very wide gain range, to employnegative feedback, to calculate measurements precisely at very lowamplitude levels, and to compensate for dynamic effects such as rate ofchange of amplitude and flowtube dynamics, and also may stem from themeter's use of a precise digital amplitude control algorithm.

The digital flowmeter detects the onset of aeration when the requireddriver gain rises simultaneously with a drop in apparent fluid density.The digital flowmeter then may directly respond to detected aeration. Ingeneral, the meter monitors the presence of aeration by comparing theobserved density of the material flowing through the conduit (i.e., thedensity measurement obtained through normal measurement techniques) tothe known, nonaerated density of the material. The controller determinesthe level of aeration based on any difference between the observed andactual densities. The controller then corrects the mass flow measurementaccordingly.

The controller determines the non-aerated density of the material bymonitoring the density over time periods in which aeration is notpresent (i.e., periods in which the density has a stable value).Alternatively, a control system to which the controller is connected mayprovide the non-aerated density as an initialization parameter.

In one implementation, the controller uses three corrections to accountfor the effects of aeration: bubble effect correction, damping effectcorrection, and sensor imbalance correction. FIGS. 40A-40H illustratethe effects of the correction procedure.

FIG. 40A illustrates the error in the phase measurement as the measureddensity decreases (i.e., as aeration increases) for different mass flowrates, absent aeration correction. As shown, the phase error is negativeand has a magnitude that increases with increasing aeration. FIG. 40Billustrates that the resulting mass flow error is also negative. It alsois significant to note that the digital flowmeter operates at highlevels of aeration. By contrast, as indicated by the horizontal bar4000, traditional analog meters tend to stall in the presence of lowlevels of aeration.

A stall occurs when the flowmeter is unable to provide a sufficientlylarge driver gain to allow high drive current at low amplitudes ofoscillation. If the level of damping requires a higher driver gain thancan be delivered by the flowtube in order to maintain oscillation at acertain amplitude, then insufficient drive energy is supplied to theconduit. This results in a drop in amplitude of oscillation, which inturn leads to even less drive energy supplied due to the maximum gainlimit. Catastrophic collapse results, and flowtube oscillation is notpossible until the damping reduces to a level at which the correspondingdriver gain requirement can be supplied by the flowmeter.

The bubble effect correction is based on the assumption that the massflow decreases as the level of aeration, also referred to as the voidfraction, increases. Without attempting to predict the actualrelationship between void fraction and the bubble effect, thiscorrection assumes, with good theoretical justification, that the effecton the observed mass flow will be the same as the effect on the observeddensity. Since the true fluid density is known, the bubble effectcorrection corrects the mass flow rate by the same proportion. Thiscorrection is a linear adjustment that is the same for all flow rates.FIGS. 40C and 40D illustrate, respectively, the residual phase and massflow errors after correction for the bubble effect. As shown, theresidual errors are now positive and substantially smaller in magnitudethan the original errors.

The damping factor correction accounts for damping of the conduit motiondue to aeration. In general, the damping factor correction is based onthe following relationship between the observed phase, φ_(obs), and theactual phase, φ_(true):

${\varphi_{obs} = {\varphi_{true}( {1 + \frac{k\;\lambda^{2}\varphi_{true}}{f^{2}}} )}},$

where Î is a damping coefficient and k is a constant. FIG. 40Eillustrates the damping correction for different mass flow rates anddifferent levels of aeration. FIG. 40F illustrates the residual phaseerror after correction for damping. As shown, the phase error is reducedsubstantially relative to the phase error remaining after bubble effectcorrection.

The sensor balance correction is based on density differences betweendifferent ends of the conduit. As shown in FIG. 41, a pressure dropbetween the inlet and the outlet of the conduit results in increasingbubble sizes from the inlet to the outlet. Since material flows seriallythrough the two loops of the conduit, the bubbles at the inlet side ofthe conduit (i.e., the side adjacent to the first sensor/driver pair)will be smaller than the bubbles at the outlet side of the conduit(i.e., the side adjacent to the second sensor/driver pair). Thisdifference in bubble size results in a difference in mass and densitybetween the two ends of the conduit. This difference is reflected in thesensor signals (SV₁ and SV₂). Accordingly, the sensor balance correctionis based on a ratio of the two sensor signals.

FIG. 40G illustrates the sensor balance correction for different massflow rates and different levels of aeration. FIG. 40H illustrates theresidual phase error after applying the sensor balance correction. Atlow flow rates and low levels of aeration, the phase error is improvedrelative to the phase error remaining after damping correction.

Other correction factors also could be used. For example, the phaseangle of each sensor signal could be monitored. In general, the averagephase angle for a signal should be zero. However, the average phaseangle tends to increase with increasing aeration. Accordingly, acorrection factor could be generated based on the value of the averagephase angle. Another correction factor could be based on the temperatureof the conduit.

In general, application of the correction factors tends to keep the massflow errors at one percent or less. Moreover, these correction factorsappear to be applicable over a wide range of flows and aeration levels.

J. Setpoint Adjustment

The digital flowmeter provides improved control of the setpoint for theamplitude of oscillation of the conduit. In an analog meter, feedbackcontrol is used to maintain the amplitude of oscillation of the conduitat a fixed level corresponding to a desired peak sensor voltage (e.g.,0.3 V). A stable amplitude of oscillation leads to reduced variance inthe frequency and phase measurements.

In general, a large amplitude of oscillation is desirable, since such alarge amplitude provides a large Coriolis signal for measurementpurposes. A large amplitude of oscillation also results in storage of ahigher level of energy in the conduit, which provides greater robustnessto external vibrations.

Circumstances may arise in which it is not possible to maintain thelarge amplitude of oscillation due to limitations in the current thatcan be supplied to the drivers. For example, in one implementation of ananalog transmitter, the current is limited to 100 mA for safetypurposes. This is typically 5-10 times the current needed to maintainthe desired amplitude of oscillation. However, if the process fluidprovides significant additional damping (e.g., via two-phase flow), thenthe optimal amplitude may no longer be sustainable.

Similarly, a low-power flowmeter, such as the two-wire meter describedbelow, may have much less power available to drive the conduit. Inaddition, the power level may vary when the conduit is driven bycapacitive discharge.

Referring to FIG. 42, a control procedure 4200 implemented by thecontroller of the digital flowmeter may be used to select the highestsustainable setpoint given a maximum available current level. Ingeneral, the procedure is performed each time that a desired drivecurrent output is selected, which typically is once per cycle, or onceevery half-cycle if interleaved cycles are used.

The controller starts by setting the setpoint to a default value (e.g.,0.3 V) and initializing filtered representations of the sensor voltage(filtered_SV) and the drive current (filtered_DC) (step 4205). Each timethat the procedure is performed, the controller updates the filteredvalues based on current values for the sensor voltage (SV) and drivecurrent (DC) (step 4210). For example, the controller may generate a newvalue for filtered_SV as the sum of ninety nine percent of filtered_SVand one percent of SV.

Next, the controller determines whether the procedure has been paused toprovide time for prior setpoint adjustments to take effect (step 4215).Pausing of the procedure is indicated by a pause cycle count having avalue greater than zero. If the procedure is paused, the controllerperforms no further actions for the cycle and decrements the pause cyclecount (step 4220).

If the procedure has not been paused, the controller determines whetherthe filtered drive current exceeds a threshold level (step 4225). In oneimplementation, the threshold level is ninety five percent of themaximum available current. If the current exceeds the threshold, thecontroller reduces the setpoint (step 4230). To allow time for the meterto settle after the setpoint change, the controller then implements apause of the procedure by setting the pause cycle count equal to anappropriate value (e.g., 100) (step 4235).

If the procedure has not been paused, the controller determines whetherthe filtered drive current is less than a threshold level (step 4240)and the setpoint is less than a maximum permitted setpoint (step 4245).In one implementation, the threshold level equals seventy percent of themaximum available current. If both conditions are met, the controllerdetermines a possible new setpoint (step 4250). In one implementation,the controller determines the new setpoint as eighty percent of themaximum available current multiplied by the ratio of filtered_SV tofiltered_DC. To avoid small changes in the setpoint (i.e., chattering),the controller then determines whether the possible new setpoint exceedsthe current setpoint by a sufficient amount (step 4255). In oneimplementation, the possible new setpoint must exceed the currentsetpoint by 0.02 V and by ten percent.

If the possible new setpoint is sufficiently large, the controllerdetermines if it is greater than the maximum permitted setpoint (step4260). If so, the controller sets the setpoint equal to the maximumpermitted setpoint (step 4265). Otherwise, the controller sets thesetpoint equal to the possible new setpoint (step 4270). The controllerthen implements a pause of the procedure by setting the pause cyclecount equal to an appropriate value (step 4235).

FIGS. 43A-43C illustrate operation of the setpoint adjustment procedure.As shown in FIG. 43C, the system starts with a setpoint of 0.3 V. Atabout eight seconds of operation, aeration results in a drop in theapparent density of the material in the conduit (FIG. 43A). Increaseddamping accompanying the aeration results in an increase in the drivecurrent (FIG. 43B) and increased noise in the sensor voltage (FIG. 43C).No changes are made at this time, since the meter is able to maintainthe desired setpoint.

At about fifteen seconds of operation, aeration increases and theapparent density decreases further (FIG. 43A). At this level ofaeration, the driver current (FIG. 43B) reaches a maximum value that isinsufficient to maintain the 0.3 V setpoint. Accordingly, the sensorvoltage drops to 0.26 V (FIG. 43C), the voltage level that the maximumdriver current is able to maintain. In response to this condition, thecontroller adjusts the setpoint (at about 28 seconds of operation) to alevel (0.23 V) that does not require generation of the maximum drivercurrent.

At about 38 seconds of operation, the level of aeration decreases andthe apparent density increases (FIG. 43A). This results in a decrease inthe drive current (FIG. 43B). At 40 seconds of operation, the controllerresponds to this condition by increasing the setpoint (FIG. 43C). Thelevel of aeration decreases and the apparent density increases again atabout 48 seconds of operation, and the controller responds by increasingthe setpoint to 0.3 V.

K. Performance Results

The digital flowmeter has shown remarkable performance improvementsrelative to traditional analog flowmeters. In one experiment, theability of the two types of meters to accurately measure a batch ofmaterial was examined. In each case, the batch was fed through theappropriate flowmeter and into a tank, where the batch was then weighed.For 1200 and 2400 pound batches, the analog meter provided an averageoffset of 500 pounds, with a repeatability of 200 pounds. By contrast,the digital meter provided an average offset of 40 pounds, with arepeatability of two pounds, which clearly is a substantial improvement.

In each case, the conduit and surrounding pipework were empty at thestart of the batch. This is important in many batching applicationswhere it is not practical to start the batch with the conduit full. Thebatches were finished with the flowtube full. Some positive offset isexpected because the flowmeter is measuring the material needed to fillthe pipe before the weighing tank starts to be filled. Delays instarting up, or offsets caused by aerated flow or low amplitudes ofoscillation, are likely to introduce negative offsets. For real batchingapplications, the most important issue is the repeatability of themeasurement.

The results show that with the analog flowmeter there are large negativeoffsets and repeatability of only 200 pounds. This is attributable tothe length of time taken to startup after the onset of flow (duringwhich no flow is metered), and measurement errors until full amplitudeof oscillation is achieved. By comparison, the digital flowmeterachieves a positive offset, which is attributable to filling up of theempty pipe, and a repeatability is two pounds.

Another experiment compared the general measurement accuracy of the twotypes of meters. FIG. 44 illustrates the accuracy and correspondinguncertainty of the measurements produced by the two types of meters atdifferent percentages of the meters' maximum recommended flow rate. Athigh flow rates (i.e., at rates of 25% or more of the maximum rate), theanalog meter produces measurements that correspond to actual values towithin 0.15% or less, compared to 0.005% or less for the digital meter.At lower rates, the offset of the analog meter is on the order of 1.5%,compared to 0.25% for the digital meter.

L. Self-Validating Meter

The digital flowmeter may used in a control system that includesself-validating sensors. To this end, the digital flowmeter may beimplemented as a self-validating meter. Self-validating meters and othersensors are described in U.S. Pat. No. 5,570,300, entitled“SELF-VALIDATING SENSORS”, which is incorporated by reference.

In general, a self-validating meter provides, based on all informationavailable to the meter, a best estimate of the value of a parameter(e.g., mass flow) being monitored. Because the best estimate is based,in part, on nonmeasurement data, the best estimate does not alwaysconform to the value indicated by the current, possibly faulty,measurement data. A self-validating meter also provides informationabout the uncertainty and reliability of the best estimate, as well asinformation about the operational status of the sensor. Uncertaintyinformation is derived from known uncertainty analyses and is providedeven in the absence of faults.

In general, a self-validating meter provides four basic parameters: avalidated measurement value (VMV), a validated uncertainty (VU), anindication (MV status) of the status under which the measurement wasgenerated, and a device status. The VMV is the meter's best estimate ofthe value of a measured parameter. The VU and the MV status areassociated with the VMV. The meter produces a separate VMV, VU and MVstatus for each measurement. The device status indicates the operationalstatus of the meter.

The meter also may provide other information. For example, upon arequest from a control system, the meter may provide detailed diagnosticinformation about the status of the meter. Also, when a measurement hasexceeded, or is about to exceed, a predetermined limit, the meter cansend an alarm signal to the control system. Different alarm levels canbe used to indicate the severity with which the measurement has deviatedfrom the predetermined value.

VMV and VU are numeric values. For example, VMV could be a temperaturemeasurement valued at 200 degrees and VU, the uncertainty of VMV, couldbe 9 degrees. In this case, there is a high probability (typically 95%)that the actual temperature being measured falls within an envelopearound VMV and designated by VU (i.e., from 191 degrees to 209 degrees).

The controller generates VMV based on underlying data from the sensors.First, the controller derives a raw measurement value (RMV) that isbased on the signals from the sensors. In general, when the controllerdetects no abnormalities, the controller has nominal confidence in theRMV and sets the VMV equal to the RMV. When the controller detects anabnormality in the sensor, the controller does not set the VMV equal tothe RMV. Instead, the controller sets the VMV equal to a value that thecontroller considers to be a better estimate than the RMV of the actualparameter.

The controller generates the VU based on a raw uncertainty signal (RU)that is the result of a dynamic uncertainty analysis of the RMV. Thecontroller performs this uncertainty analysis during each samplingperiod. Uncertainty analysis, originally described in “DescribingUncertainties in Single Sample Experiments,” S. J. Kline & F. A.McClintock, Mech. Eng., 75, 3-8 (1953), has been widely applied and hasachieved the status of an international standard for calibration.Essentially, an uncertainty analysis provides an indication of the“quality” of a measurement. Every measurement has an associated error,which, of course, is unknown. However, a reasonable limit on that errorcan often be expressed by a single uncertainty number (ANSI/ASME PTC19.1-1985 Part 1, Measurement Uncertainty Instruments and Apparatus).

As described by Kline & McClintock, for any observed measurement M, theuncertainty in M, w_(M), can be defined as follows:M _(true) ε[M−w _(M) ,M+w _(M)]

where M is true (M_(true)) with a certain level of confidence (typically95%). This uncertainty is readily expressed in a relative form as aproportion of the measurement (i.e. w_(M)/M).

In general, the VU has a non-zero value even under ideal conditions(i.e., a faultless sensor operating in a controlled, laboratoryenvironment). This is because the measurement produced by a sensor isnever completely certain and there is always some potential for error.As with the VMV, when the controller detects no abnormalities, thecontroller sets the VU equal to the RU. When the controller detects afault that only partially affects the reliability of the RMV, thecontroller typically performs a new uncertainty analysis that accountsfor effects of the fault and sets the VU equal to the results of thisanalysis. The controller sets the VU to a value based on pastperformance when the controller determines that the RMV bears norelation to the actual measured value.

To ensure that the control system uses the VMV and the VU properly, theMV status provides information about how they were calculated. Thecontroller produces the VMV and the VU under all conditions—even whenthe sensors are inoperative. The control system needs to know whetherVMV and VU are based on “live” or historical data. For example, if thecontrol system were using VMV and VU in feedback control and the sensorswere inoperative, the control system would need to know that VMV and VUwere based on past performance.

The MV status is based on the expected persistence of any abnormalcondition and on the confidence of the controller in the RMV. The fourprimary states for MV status are generated according to Table 1.

TABLE 1 Expected Persistence Confidence in RMV MV Status not applicablenominal CLEAR not applicable reduced BLURRED short zero DAZZLED longzero BLIND

A CLEAR MV status occurs when RMV is within a normal range for givenprocess conditions. A DAZZLED MV status indicates that RMV is quiteabnormal, but the abnormality is expected to be of short duration.Typically, the controller sets the MV status to DAZZLED when there is asudden change in the signal from one of the sensors and the controlleris unable to clearly establish whether this change is due to an as yetundiagnosed sensor fault or to an abrupt change in the variable beingmeasured. A BLURRED MV status indicates that the RMV is abnormal butreasonably related to the parameter being measured. For example, thecontroller may set the MV status to BLURRED when the RMV is a noisysignal. A BLIND MV status indicates that the RMV is completelyunreliable and that the fault is expected to persist.

Two additional states for the MV status are UNVALIDATED and SECURE. TheMV status is UNVALIDATED when the controller is not performingvalidation of VMV. MV status is SECURE when VMV is generated fromredundant measurements in which the controller has nominal confidence.

The device status is a generic, discrete value summarizing the health ofthe meter. It is used primarily by fault detection and maintenanceroutines of the control system. Typically, the device status 32 is inone of six states, each of which indicates a different operationalstatus for the meter. These states are: GOOD, TESTING, SUSPECT,IMPAIRED, BAD, or CRITICAL. A GOOD device status means that the meter isin nominal condition. A TESTING device status means that the meter isperforming a self check, and that this self check may be responsible forany temporary reduction in measurement quality. A SUSPECT device statusmeans that the meter has produced an abnormal response, but thecontroller has no detailed fault diagnosis. An IMPAIRED device statusmeans that the meter is suffering from a diagnosed fault that has aminor impact on performance. A BAD device status means that the meterhas seriously malfunctioned and maintenance is required. Finally, aCRITICAL device status means that the meter has malfunctioned to theextent that the meter may cause (or have caused) a hazard such as aleak, fire, or explosion.

FIG. 45 illustrates a procedure 4500 by which the controller of aself-validating meter processes digitized sensor signals to generatedrive signals and a validated mass flow measurement with an accompanyinguncertainty and measurement status. Initially, the controller collectsdata from the sensors (step 4505). Using this data, the controllerdetermines the frequency of the sensor signals (step 4510). If thefrequency falls within an expected range (step 4515), the controllereliminates zero offset from the sensor signals (step 4520), anddetermines the amplitude (step 4525) and phase (step 4530) of the sensorsignals. The controller uses these calculated values to generate thedrive signal (step 4535) and to generate a raw mass flow measurement andother measurements (step 4540).

If the frequency does not fall within an expected range (step 4515),then the controller implements a stall procedure (step 4545) todetermine whether the conduit has stalled and to respond accordingly. Inthe stall procedure, the controller maximizes the driver gain andperforms a broader search for zero crossings to determine whether theconduit is oscillating at all.

If the conduit is not oscillating correctly (i.e., if it is notoscillating, or if it is oscillating at an unacceptably high frequency(e.g., at a high harmonic of the resonant frequency)) (step 4550), thecontroller attempts to restart normal oscillation (step 4555) of theconduit by, for example, injecting a square wave at the drivers. Afterattempting to restart oscillation, the controller sets the MV status toDAZZLED (step 4560) and generates null raw measurement values (step4565). If the conduit is oscillating correctly (step 4550), thecontroller eliminates zero offset (step 4520) and proceeds as discussedabove.

After generating raw measurement values (steps 4540 or 4565), thecontroller performs diagnostics (step 4570) to determine whether themeter is operating correctly (step 4575). (Note that the controller doesnot necessarily perform these diagnostics during every cycle.)

Next, the controller performs an uncertainty analysis (step 4580) togenerate a raw uncertainty value. Using the raw measurements, theresults of the diagnostics, and other information, the controllergenerates the VMV, the VU, the MV status, and the device status (step4585). Thereafter, the controller collects a new set of data and repeatsthe procedure. The steps of the procedure 4500 may be performed seriallyor in parallel, and may be performed in varying order.

In another example, when aeration is detected, the mass flow correctionsare applied as described above, the MV status becomes blurred, and theuncertainty is increased to reflect the probable error of the correctiontechnique. For example, for a flowtube operating at 50% flowrate, undernormal operating conditions, the uncertainty might be of the order of0.1-0.2% of flowrate. If aeration occurs and is corrected for using thetechniques described above, the uncertainty might be increased toperhaps 2% of reading. Uncertainty values should decrease asunderstanding of the effects of aeration improves and the ability tocompensate for aeration gets better. In batch situations, where flowrate uncertainty is variable (e.g. high at start/end if batching from/toempty, or during temporary incidents of aeration or cavitation), theuncertainty of the batch total will reflect the weighted significance ofthe periods of high uncertainty against the rest of the batch withnominal low uncertainty. This is a highly useful quality metric infiscal and other metering applications.

M. Two Wire Flowmeter

Other embodiments are also contemplated. For example, as shown in FIG.46, the techniques described above may be used to implement a “two-wire”Coriolis flowmeter 4600 that performs bidirectional communications on apair of wires 4605. A power circuit 4610 receives power for operating adigital controller 4615 and for powering the driver(s) 4620 to vibratethe conduit 4625. For example, the power circuit may include a constantoutput circuit 4630 that provides operating power to the controller anda drive capacitor 4635 that is charged using excess power. The powercircuit may receive power from the wires 4605 or from a second pair ofwires. The digital controller receives signals from one or more sensors4640.

When the drive capacitor is suitably charged, the controller 4615discharges the capacitor 4635 to drive the conduit 4625. For example,the controller may drive the conduit once during every 10 cycles. Thecontroller 4615 receives and analyzes signals from the sensors 4640 toproduce a mass flow measurement that the controller then transmits onthe wires 4605.

A number of implementations have been described. Nevertheless, it willbe understood that various modifications may be made. Accordingly, otherimplementations are within the scope of the following claimsapplication.

What is claimed is:
 1. A Coriolis flowmeter comprising: a pair ofconduits configured to convey a fluid through the flowmeter, eachconduit defining a fluid passage having at least one curve; a driverconfigured to oscillate the conduits in substantially opposite phase toone another; a first sensor configured to generate a first signalindicative of relative movement of the pair of conduits when they areoscillated by the driver, the first sensor being positioned at a firstlocation; a second sensor configured to generate a second signalindicative of relative movement of the pair of conduits when they areoscillated by the driver, the second sensor being positioned at a secondlocation, the first and second sensors being positioned so a phasedifference between the first and second signals is related to a massflow rate of the fluid through the flowmeter; a digital signal processorconfigured to detect the phase difference and determine the mass flowrate of the fluid using the detected phase difference and output asignal indicative of the determined mass flow rate; and a digital drivesignal generator configured to: combine the first and second signals toproduce a combined sensor signal; produce a variable digital drivesignal using the combined sensor signal, the variable digital drivesignal having a frequency that is substantially at a resonant frequencyof the conduits as determined using at least one of the first and secondsignals; determine a time delay associated with one or more componentsconnected between the sensors and the driver; adjust a phase of thevariable digital drive signal to compensate for the determined timedelay; and selectively adjust the drive signal to cause the driver toresist motion of the pair of conduits when an amplitude of at least oneof the sensor signals exceeds a set point.
 2. A Coriolis flowmeter asset forth in claim 1 wherein the flowmeter is configured to determinethe mass flow rate of the fluid using a correction to account foraeration.
 3. A Coriolis flowmeter as set forth in claim 1 wherein thefirst and second sensor signals are generally periodic and the digitalsignal processor processes data from the first and second sensor signalsin sets, each set including data for a complete cycle of the respectivesignal.
 4. A Coriolis flowmeter as set forth in claim 3 wherein thedigital signal processor is configured to allow the number of samples inthe sets to vary.